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Theorem List for Metamath Proof Explorer - 1601-1700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremalnex 1601 Theorem 19.7 of [Margaris] p. 89. (Contributed by NM, 12-Mar-1993.)
 |-  ( A. x  -.  ph  <->  -. 
 E. x ph )
 
Theoremeximal 1602 A utility theorem. An interesting case is when the same formula is substituted for both  ph and  ps, since then both implications express a type of non-freeness. See also alimex 1639. (Contributed by BJ, 12-May-2019.)
 |-  ( ( E. x ph 
 ->  ps )  <->  ( -.  ps  ->  A. x  -.  ph ) )
 
Syntaxwnf 1603 Extend wff definition to include the not-free predicate.
 wff  F/ x ph
 
Definitiondf-nf 1604 Define the not-free predicate for wffs. This is read " x is not free in  ph". Not-free means that the value of  x cannot affect the value of  ph, e.g., any occurrence of  x in  ph is effectively bound by a "for all" or something that expands to one (such as "there exists"). In particular, substitution for a variable not free in a wff does not affect its value (sbf 2107). An example of where this is used is stdpc5 1894. See nf2 1946 for an alternative definition which does not involve nested quantifiers on the same variable.

Not-free is a commonly used constraint, so it is useful to have a notation for it. Surprisingly, there is no common formal notation for it, so here we devise one. Our definition lets us work with the not-free notion within the logic itself rather than as a metalogical side condition.

To be precise, our definition really means "effectively not free," because it is slightly less restrictive than the usual textbook definition for not-free (which only considers syntactic freedom). For example,  x is effectively not free in the bare expression  x  =  x (see nfequid 1778), even though  x would be considered free in the usual textbook definition, because the value of  x in the expression  x  =  x cannot affect the truth of the expression (and thus substitution will not change the result).

This predicate only applies to wffs. See df-nfc 2593 for a not-free predicate for class variables. (Contributed by Mario Carneiro, 11-Aug-2016.)

 |-  ( F/ x ph  <->  A. x ( ph  ->  A. x ph ) )
 
1.4.2  Rule scheme ax-gen (Generalization)
 
Axiomax-gen 1605 Rule of Generalization. The postulated inference rule of predicate calculus. See e.g. Rule 2 of [Hamilton] p. 74. This rule says that if something is unconditionally true, then it is true for all values of a variable. For example, if we have proved  x  =  x, we can conclude  A. x x  =  x or even  A. y
x  =  x. Theorem allt 29842 shows the special case  A. x T.. Theorem spi 1850 shows we can go the other way also: in other words we can add or remove universal quantifiers from the beginning of any theorem as required. (Contributed by NM, 3-Jan-1993.)
 |-  ph   =>    |- 
 A. x ph
 
Theoremgen2 1606 Generalization applied twice. (Contributed by NM, 30-Apr-1998.)
 |-  ph   =>    |- 
 A. x A. y ph
 
Theoremmpg 1607 Modus ponens combined with generalization. (Contributed by NM, 24-May-1994.)
 |-  ( A. x ph  ->  ps )   &    |-  ph   =>    |- 
 ps
 
Theoremmpgbi 1608 Modus ponens on biconditional combined with generalization. (Contributed by NM, 24-May-1994.) (Proof shortened by Stefan Allan, 28-Oct-2008.)
 |-  ( A. x ph  <->  ps )   &    |-  ph   =>    |- 
 ps
 
Theoremmpgbir 1609 Modus ponens on biconditional combined with generalization. (Contributed by NM, 24-May-1994.) (Proof shortened by Stefan Allan, 28-Oct-2008.)
 |-  ( ph  <->  A. x ps )   &    |-  ps   =>    |-  ph
 
Theoremnfi 1610 Deduce that  x is not free in  ph from the definition. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |-  ( ph  ->  A. x ph )   =>    |- 
 F/ x ph
 
Theoremhbth 1611 No variable is (effectively) free in a theorem.

This and later "hypothesis-building" lemmas, with labels starting "hb...", allow us to construct proofs of formulas of the form  |-  ( ph  ->  A. x ph ) from smaller formulas of this form. These are useful for constructing hypotheses that state " x is (effectively) not free in  ph." (Contributed by NM, 11-May-1993.)

 |-  ph   =>    |-  ( ph  ->  A. x ph )
 
Theoremnfth 1612 No variable is (effectively) free in a theorem. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |-  ph   =>    |- 
 F/ x ph
 
Theoremnftru 1613 The true constant has no free variables. (This can also be proven in one step with nfv 1694, but this proof does not use ax-5 1691.) (Contributed by Mario Carneiro, 6-Oct-2016.)
 |- 
 F/ x T.
 
Theoremnex 1614 Generalization rule for negated wff. (Contributed by NM, 18-May-1994.)
 |- 
 -.  ph   =>    |- 
 -.  E. x ph
 
Theoremnfnth 1615 No variable is (effectively) free in a non-theorem. (Contributed by Mario Carneiro, 6-Dec-2016.)
 |- 
 -.  ph   =>    |- 
 F/ x ph
 
Theoremnffal 1616 The false constant has no free variables (see nftru 1613). (Contributed by BJ, 6-May-2019.)
 |- 
 F/ x F.
 
Theoremsptruw 1617 Version of sp 1845 when  ph is true. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 23-Apr-2017.)
 |-  ph   =>    |-  ( A. x ph  -> 
 ph )
 
1.4.3  Axiom scheme ax-4 (Quantified Implication)
 
Axiomax-4 1618 Axiom of Quantified Implication. Axiom C4 of [Monk2] p. 105 and Theorem 19.20 of [Margaris] p. 90. It is restated as alim 1619 for labelling consistency. It should be used only by alim 1619. (Contributed by NM, 21-May-2008.) (New usage is discouraged.)
 |-  ( A. x (
 ph  ->  ps )  ->  ( A. x ph  ->  A. x ps ) )
 
Theoremalim 1619 Restatement of Axiom ax-4 1618, for labelling consistency. It should be the only theorem using ax-4 1618. (Contributed by NM, 10-Jan-1993.)
 |-  ( A. x (
 ph  ->  ps )  ->  ( A. x ph  ->  A. x ps ) )
 
Theoremalimi 1620 Inference quantifying both antecedent and consequent. (Contributed by NM, 5-Jan-1993.)
 |-  ( ph  ->  ps )   =>    |-  ( A. x ph  ->  A. x ps )
 
Theorem2alimi 1621 Inference doubly quantifying both antecedent and consequent. (Contributed by NM, 3-Feb-2005.)
 |-  ( ph  ->  ps )   =>    |-  ( A. x A. y ph  ->  A. x A. y ps )
 
Theoremal2im 1622 Closed form of al2imi 1623. Version of ax-4 1618 for a nested implication. (Contributed by Alan Sare, 31-Dec-2011.)
 |-  ( A. x (
 ph  ->  ( ps  ->  ch ) )  ->  ( A. x ph  ->  ( A. x ps  ->  A. x ch ) ) )
 
Theoremal2imi 1623 Inference quantifying antecedent, nested antecedent, and consequent. (Contributed by NM, 10-Jan-1993.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( A. x ph 
 ->  ( A. x ps  ->  A. x ch )
 )
 
Theoremalanimi 1624 Variant of al2imi 1623 with conjunctive antecedent. (Contributed by Andrew Salmon, 8-Jun-2011.)
 |-  ( ( ph  /\  ps )  ->  ch )   =>    |-  ( ( A. x ph 
 /\  A. x ps )  ->  A. x ch )
 
Theoremalimdh 1625 Deduction form of Theorem 19.20 of [Margaris] p. 90, see alim 1619. (Contributed by NM, 4-Jan-2002.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ph  ->  ( ps  ->  ch )
 )   =>    |-  ( ph  ->  ( A. x ps  ->  A. x ch ) )
 
Theoremalbi 1626 Theorem 19.15 of [Margaris] p. 90. (Contributed by NM, 24-Jan-1993.)
 |-  ( A. x (
 ph 
 <->  ps )  ->  ( A. x ph  <->  A. x ps )
 )
 
Theoremalbii 1627 Inference adding universal quantifier to both sides of an equivalence. (Contributed by NM, 7-Aug-1994.)
 |-  ( ph  <->  ps )   =>    |-  ( A. x ph  <->  A. x ps )
 
Theorem2albii 1628 Inference adding two universal quantifiers to both sides of an equivalence. (Contributed by NM, 9-Mar-1997.)
 |-  ( ph  <->  ps )   =>    |-  ( A. x A. y ph  <->  A. x A. y ps )
 
Theoremalrimih 1629 Inference form of Theorem 19.21 of [Margaris] p. 90. See 19.21 1891 and 19.21h 1893. (Contributed by NM, 9-Jan-1993.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ph  ->  ps )   =>    |-  ( ph  ->  A. x ps )
 
Theoremhbxfrbi 1630 A utility lemma to transfer a bound-variable hypothesis builder into a definition. See hbxfreq 2565 for equality version. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
 |-  ( ph  <->  ps )   &    |-  ( ps  ->  A. x ps )   =>    |-  ( ph  ->  A. x ph )
 
Theoremnfbii 1631 Equality theorem for not-free. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |-  ( ph  <->  ps )   =>    |-  ( F/ x ph  <->  F/ x ps )
 
Theoremnfxfr 1632 A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |-  ( ph  <->  ps )   &    |-  F/ x ps   =>    |-  F/ x ph
 
Theoremnfxfrd 1633 A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Mario Carneiro, 24-Sep-2016.)
 |-  ( ph  <->  ps )   &    |-  ( ch  ->  F/ x ps )   =>    |-  ( ch  ->  F/ x ph )
 
Theoremalex 1634 Theorem 19.6 of [Margaris] p. 89. (Contributed by NM, 12-Mar-1993.)
 |-  ( A. x ph  <->  -.  E. x  -.  ph )
 
Theoremexnal 1635 Theorem 19.14 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
 |-  ( E. x  -.  ph  <->  -. 
 A. x ph )
 
Theorem2nalexn 1636 Part of theorem *11.5 in [WhiteheadRussell] p. 164. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( -.  A. x A. y ph  <->  E. x E. y  -.  ph )
 
Theorem2exnaln 1637 Theorem *11.22 in [WhiteheadRussell] p. 160. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( E. x E. y ph  <->  -.  A. x A. y  -.  ph )
 
Theorem2nexaln 1638 Theorem *11.25 in [WhiteheadRussell] p. 160. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( -.  E. x E. y ph  <->  A. x A. y  -.  ph )
 
Theoremalimex 1639 A utility theorem. An interesting case is when the same formula is substituted for both  ph and  ps, since then both implications express a type of non-freeness. See also eximal 1602. (Contributed by BJ, 12-May-2019.)
 |-  ( ( ph  ->  A. x ps )  <->  ( E. x  -.  ps  ->  -.  ph )
 )
 
Theoremaleximi 1640 A variant of al2imi 1623: instead of applying  A. x quantifiers to the final implication, replace them with  E. x. A shorter proof is possible using nfa1 1883, sps 1851 and eximd 1868, but it depends on more axioms. (Contributed by Wolf Lammen, 18-Aug-2019.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( A. x ph 
 ->  ( E. x ps  ->  E. x ch )
 )
 
Theoremexim 1641 Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 10-Jan-1993.) (Proof shortened by Wolf Lammen, 4-Jul-2014.)
 |-  ( A. x (
 ph  ->  ps )  ->  ( E. x ph  ->  E. x ps ) )
 
TheoremeximOLD 1642 Obsolete proof of exim 1641 as of 4-Sep-2019. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 4-Jul-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A. x (
 ph  ->  ps )  ->  ( E. x ph  ->  E. x ps ) )
 
Theoremeximi 1643 Inference adding existential quantifier to antecedent and consequent. (Contributed by NM, 10-Jan-1993.)
 |-  ( ph  ->  ps )   =>    |-  ( E. x ph  ->  E. x ps )
 
Theorem2eximi 1644 Inference adding two existential quantifiers to antecedent and consequent. (Contributed by NM, 3-Feb-2005.)
 |-  ( ph  ->  ps )   =>    |-  ( E. x E. y ph  ->  E. x E. y ps )
 
Theoremeximii 1645 Inference associated with eximi 1643. (Contributed by BJ, 3-Feb-2018.)
 |- 
 E. x ph   &    |-  ( ph  ->  ps )   =>    |- 
 E. x ps
 
TheoremaleximiOLD 1646 Obsolete proof of aleximi 1640 as of 4-Sep-2019. (Contributed by Wolf Lammen, 18-Aug-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( A. x ph 
 ->  ( E. x ps  ->  E. x ch )
 )
 
Theoremala1 1647 Add an antecedent in a universally quantified formula. (Contributed by BJ, 6-Oct-2018.)
 |-  ( A. x ph  ->  A. x ( ps 
 ->  ph ) )
 
Theoremexa1 1648 Add an antecedent in an existentially quantified formula. (Contributed by BJ, 6-Oct-2018.)
 |-  ( E. x ph  ->  E. x ( ps 
 ->  ph ) )
 
Theorem19.38 1649 Theorem 19.38 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.) Allow a shortening of 19.21t 1890 and 19.23t 1895. (Revised by Wolf Lammen, 2-Jan-2018.)
 |-  ( ( E. x ph 
 ->  A. x ps )  ->  A. x ( ph  ->  ps ) )
 
Theoremalinexa 1650 A transformation of quantifiers and logical connectives. (Contributed by NM, 19-Aug-1993.)
 |-  ( A. x (
 ph  ->  -.  ps )  <->  -. 
 E. x ( ph  /\ 
 ps ) )
 
Theoremalexn 1651 A relationship between two quantifiers and negation. (Contributed by NM, 18-Aug-1993.)
 |-  ( A. x E. y  -.  ph  <->  -.  E. x A. y ph )
 
Theorem2exnexn 1652 Theorem *11.51 in [WhiteheadRussell] p. 164. (Contributed by Andrew Salmon, 24-May-2011.) (Proof shortened by Wolf Lammen, 25-Sep-2014.)
 |-  ( E. x A. y ph  <->  -.  A. x E. y  -.  ph )
 
Theoremexbi 1653 Theorem 19.18 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
 |-  ( A. x (
 ph 
 <->  ps )  ->  ( E. x ph  <->  E. x ps )
 )
 
Theoremexbii 1654 Inference adding existential quantifier to both sides of an equivalence. (Contributed by NM, 24-May-1994.)
 |-  ( ph  <->  ps )   =>    |-  ( E. x ph  <->  E. x ps )
 
Theorem2exbii 1655 Inference adding two existential quantifiers to both sides of an equivalence. (Contributed by NM, 16-Mar-1995.)
 |-  ( ph  <->  ps )   =>    |-  ( E. x E. y ph  <->  E. x E. y ps )
 
Theorem3exbii 1656 Inference adding 3 existential quantifiers to both sides of an equivalence. (Contributed by NM, 2-May-1995.)
 |-  ( ph  <->  ps )   =>    |-  ( E. x E. y E. z ph  <->  E. x E. y E. z ps )
 
Theoremexanali 1657 A transformation of quantifiers and logical connectives. (Contributed by NM, 25-Mar-1996.) (Proof shortened by Wolf Lammen, 4-Sep-2014.)
 |-  ( E. x (
 ph  /\  -.  ps )  <->  -. 
 A. x ( ph  ->  ps ) )
 
Theoremexancom 1658 Commutation of conjunction inside an existential quantifier. (Contributed by NM, 18-Aug-1993.)
 |-  ( E. x (
 ph  /\  ps )  <->  E. x ( ps  /\  ph ) )
 
Theoremalrimdh 1659 Deduction form of Theorem 19.21 of [Margaris] p. 90, see 19.21 1891 and 19.21h 1893. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 13-May-2011.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ps  ->  A. x ps )   &    |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( ps  ->  A. x ch )
 )
 
Theoremeximdh 1660 Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 20-May-1996.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ph  ->  ( ps  ->  ch )
 )   =>    |-  ( ph  ->  ( E. x ps  ->  E. x ch ) )
 
Theoremnexdh 1661 Deduction for generalization rule for negated wff. (Contributed by NM, 2-Jan-2002.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ph  ->  -. 
 ps )   =>    |-  ( ph  ->  -.  E. x ps )
 
Theoremalbidh 1662 Formula-building rule for universal quantifier (deduction rule). (Contributed by NM, 26-May-1993.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ph  ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  (
 A. x ps  <->  A. x ch )
 )
 
Theoremexbidh 1663 Formula-building rule for existential quantifier (deduction rule). (Contributed by NM, 26-May-1993.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ph  ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  ( E. x ps  <->  E. x ch )
 )
 
Theoremexsimpl 1664 Simplification of an existentially quantified conjunction. (Contributed by Rodolfo Medina, 25-Sep-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
 |-  ( E. x (
 ph  /\  ps )  ->  E. x ph )
 
Theoremexsimpr 1665 Simplification of an existentially quantified conjunction. (Contributed by Rodolfo Medina, 25-Sep-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
 |-  ( E. x (
 ph  /\  ps )  ->  E. x ps )
 
Theorem19.40 1666 Theorem 19.40 of [Margaris] p. 90. (Contributed by NM, 26-May-1993.)
 |-  ( E. x (
 ph  /\  ps )  ->  ( E. x ph  /\ 
 E. x ps )
 )
 
Theorem19.26 1667 Theorem 19.26 of [Margaris] p. 90. Also Theorem *10.22 of [WhiteheadRussell] p. 147. (Contributed by NM, 12-Mar-1993.) (Proof shortened by Wolf Lammen, 4-Jul-2014.)
 |-  ( A. x (
 ph  /\  ps )  <->  (
 A. x ph  /\  A. x ps ) )
 
Theorem19.26-2 1668 Theorem 19.26 1667 with two quantifiers. (Contributed by NM, 3-Feb-2005.)
 |-  ( A. x A. y ( ph  /\  ps ) 
 <->  ( A. x A. y ph  /\  A. x A. y ps ) )
 
Theorem19.26-3an 1669 Theorem 19.26 1667 with triple conjunction. (Contributed by NM, 13-Sep-2011.)
 |-  ( A. x (
 ph  /\  ps  /\  ch ) 
 <->  ( A. x ph  /\ 
 A. x ps  /\  A. x ch ) )
 
Theorem19.29 1670 Theorem 19.29 of [Margaris] p. 90. See also 19.29r 1671. (Contributed by NM, 21-Jun-1993.) (Proof shortened by Andrew Salmon, 13-May-2011.)
 |-  ( ( A. x ph 
 /\  E. x ps )  ->  E. x ( ph  /\ 
 ps ) )
 
Theorem19.29r 1671 Variation of 19.29 1670. (Contributed by NM, 18-Aug-1993.)
 |-  ( ( E. x ph 
 /\  A. x ps )  ->  E. x ( ph  /\ 
 ps ) )
 
Theorem19.29r2 1672 Variation of 19.29r 1671 with double quantification. (Contributed by NM, 3-Feb-2005.)
 |-  ( ( E. x E. y ph  /\  A. x A. y ps )  ->  E. x E. y
 ( ph  /\  ps )
 )
 
Theorem19.29x 1673 Variation of 19.29 1670 with mixed quantification. (Contributed by NM, 11-Feb-2005.)
 |-  ( ( E. x A. y ph  /\  A. x E. y ps )  ->  E. x E. y
 ( ph  /\  ps )
 )
 
Theorem19.35 1674 Theorem 19.35 of [Margaris] p. 90. This theorem is useful for moving an implication (in the form of the right-hand side) into the scope of a single existential quantifier. (Contributed by NM, 12-Mar-1993.) (Proof shortened by Wolf Lammen, 27-Jun-2014.)
 |-  ( E. x (
 ph  ->  ps )  <->  ( A. x ph 
 ->  E. x ps )
 )
 
Theorem19.35OLD 1675 Obsolete proof of 19.35 1674 as of 4-Sep-2019. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 27-Jun-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( E. x (
 ph  ->  ps )  <->  ( A. x ph 
 ->  E. x ps )
 )
 
Theorem19.35i 1676 Inference associated with 19.35 1674. (Contributed by NM, 21-Jun-1993.)
 |- 
 E. x ( ph  ->  ps )   =>    |-  ( A. x ph  ->  E. x ps )
 
Theorem19.35ri 1677 Inference associated with 19.35 1674. (Contributed by NM, 12-Mar-1993.)
 |-  ( A. x ph  ->  E. x ps )   =>    |-  E. x ( ph  ->  ps )
 
Theorem19.25 1678 Theorem 19.25 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
 |-  ( A. y E. x ( ph  ->  ps )  ->  ( E. y A. x ph  ->  E. y E. x ps ) )
 
Theorem19.30 1679 Theorem 19.30 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  ( A. x (
 ph  \/  ps )  ->  ( A. x ph  \/  E. x ps )
 )
 
Theorem19.43 1680 Theorem 19.43 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.) (Proof shortened by Wolf Lammen, 27-Jun-2014.)
 |-  ( E. x (
 ph  \/  ps )  <->  ( E. x ph  \/  E. x ps ) )
 
Theorem19.43OLD 1681 Obsolete proof of 19.43 1680 as of 3-May-2099. Leave this in for the example on the mmrecent.html page and in conventions 25101. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( E. x (
 ph  \/  ps )  <->  ( E. x ph  \/  E. x ps ) )
 
Theorem19.33 1682 Theorem 19.33 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
 |-  ( ( A. x ph 
 \/  A. x ps )  ->  A. x ( ph  \/  ps ) )
 
Theorem19.33b 1683 The antecedent provides a condition implying the converse of 19.33 1682. (Contributed by NM, 27-Mar-2004.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 5-Jul-2014.)
 |-  ( -.  ( E. x ph  /\  E. x ps )  ->  ( A. x ( ph  \/  ps )  <->  ( A. x ph 
 \/  A. x ps )
 ) )
 
Theorem19.40-2 1684 Theorem *11.42 in [WhiteheadRussell] p. 163. Theorem 19.40 of [Margaris] p. 90 with two quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( E. x E. y ( ph  /\  ps )  ->  ( E. x E. y ph  /\  E. x E. y ps )
 )
 
Theorem19.40b 1685 The antecedent provides a condition implying the converse of 19.40 1666. This is to 19.40 1666 what 19.33b 1683 is to 19.33 1682. (Contributed by BJ, 6-May-2019.)
 |-  ( ( A. x ph 
 \/  A. x ps )  ->  ( ( E. x ph 
 /\  E. x ps )  <->  E. x ( ph  /\  ps ) ) )
 
Theoremalbiim 1686 Split a biconditional and distribute quantifier. (Contributed by NM, 18-Aug-1993.)
 |-  ( A. x (
 ph 
 <->  ps )  <->  ( A. x ( ph  ->  ps )  /\  A. x ( ps 
 ->  ph ) ) )
 
Theorem2albiim 1687 Split a biconditional and distribute two quantifiers. (Contributed by NM, 3-Feb-2005.)
 |-  ( A. x A. y ( ph  <->  ps )  <->  ( A. x A. y ( ph  ->  ps )  /\  A. x A. y ( ps  ->  ph ) ) )
 
Theoremexintrbi 1688 Add/remove a conjunct in the scope of an existential quantifier. (Contributed by Raph Levien, 3-Jul-2006.)
 |-  ( A. x (
 ph  ->  ps )  ->  ( E. x ph  <->  E. x ( ph  /\ 
 ps ) ) )
 
Theoremexintr 1689 Introduce a conjunct in the scope of an existential quantifier. (Contributed by NM, 11-Aug-1993.)
 |-  ( A. x (
 ph  ->  ps )  ->  ( E. x ph  ->  E. x ( ph  /\  ps )
 ) )
 
Theoremalsyl 1690 Theorem *10.3 in [WhiteheadRussell] p. 150. (Contributed by Andrew Salmon, 8-Jun-2011.)
 |-  ( ( A. x ( ph  ->  ps )  /\  A. x ( ps 
 ->  ch ) )  ->  A. x ( ph  ->  ch ) )
 
1.4.4  Axiom scheme ax-5 (Distinctness) - first use of $d
 
Axiomax-5 1691* Axiom of Distinctness. This axiom quantifies a variable over a formula in which it does not occur. Axiom C5 in [Megill] p. 444 (p. 11 of the preprint). Also appears as Axiom B6 (p. 75) of system S2 of [Tarski] p. 77 and Axiom C5-1 of [Monk2] p. 113.

(See comments in ax5ALT 2222 about the logical redundancy of ax-5 1691 in the presence of our obsolete axioms.)

This axiom essentially says that if  x does not occur in  ph, i.e.  ph does not depend on  x in any way, then we can add the quantifier  A. x to  ph with no further assumptions. By sp 1845, we can also remove the quantifier (unconditionally). (Contributed by NM, 10-Jan-1993.)

 |-  ( ph  ->  A. x ph )
 
Theoremax5d 1692* ax-5 1691 with antecedent. Useful in proofs of deduction versions of bound-variable hypothesis builders. (Contributed by NM, 1-Mar-2013.)
 |-  ( ph  ->  ( ps  ->  A. x ps )
 )
 
Theoremax5e 1693* A rephrasing of ax-5 1691 using the existential quantifier. (Contributed by Wolf Lammen, 4-Dec-2017.)
 |-  ( E. x ph  -> 
 ph )
 
Theoremnfv 1694* If  x is not present in  ph, then  x is not free in  ph. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- 
 F/ x ph
 
Theoremnfvd 1695* nfv 1694 with antecedent. Useful in proofs of deduction versions of bound-variable hypothesis builders such as nfimd 1903. (Contributed by Mario Carneiro, 6-Oct-2016.)
 |-  ( ph  ->  F/ x ps )
 
Theoremalimdv 1696* Deduction form of Theorem 19.20 of [Margaris] p. 90, see alim 1619. (Contributed by NM, 3-Apr-1994.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  (
 A. x ps  ->  A. x ch ) )
 
Theoremeximdv 1697* Deduction form of Theorem 19.22 of [Margaris] p. 90, see exim 1641. (Contributed by NM, 27-Apr-1994.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( E. x ps  ->  E. x ch ) )
 
Theorem2alimdv 1698* Deduction form of Theorem 19.20 of [Margaris] p. 90 with two quantifiers, see alim 1619. (Contributed by NM, 27-Apr-2004.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  (
 A. x A. y ps  ->  A. x A. y ch ) )
 
Theorem2eximdv 1699* Deduction form of Theorem 19.22 of [Margaris] p. 90 with two quantifiers, see exim 1641. (Contributed by NM, 3-Aug-1995.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( E. x E. y ps  ->  E. x E. y ch ) )
 
Theoremalbidv 1700* Formula-building rule for universal quantifier (deduction rule). (Contributed by NM, 26-May-1993.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  (
 A. x ps  <->  A. x ch )
 )
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