Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  equs45f Structured version   Unicode version

Theorem equs45f 2144
 Description: Two ways of expressing substitution when is not free in . The implication "to the left" is equs4 2088 and does not require the non-freeness hypothesis. Theorem sb56 2223 replaces the non-freeness hypothesis with a dv condition. (Contributed by NM, 25-Apr-2008.) (Revised by Mario Carneiro, 4-Oct-2016.)
Hypothesis
Ref Expression
equs45f.1
Assertion
Ref Expression
equs45f

Proof of Theorem equs45f
StepHypRef Expression
1 equs45f.1 . . . . . 6
21nfri 1925 . . . . 5
32anim2i 571 . . . 4
43eximi 1702 . . 3
5 equs5a 2033 . . 3
64, 5syl 17 . 2
7 equs4 2088 . 2
86, 7impbii 190 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 187   wa 370  wal 1435  wex 1659  wnf 1663 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-12 1905  ax-13 2053 This theorem depends on definitions:  df-bi 188  df-an 372  df-ex 1660  df-nf 1664 This theorem is referenced by:  sb5f  2180
 Copyright terms: Public domain W3C validator