Theorem sb6 2268

Description: Equivalence for substitution. Compare Theorem 6.2 of [Quine] p. 40. Also proved as Lemmas 16 and 17 of [Tarski] p. 70. The implication "to the left" is sb2 2193 and does not require any dv condition. Theorem sb6f 2224 replaces the dv condition with a non-freeness hypothesis. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Wolf Lammen, 21-Sep-2018.)

Assertion

Ref	Expression
sb6	|- ( [ y / x ] ph <-> A. x ( x = y -> ph ) )

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y)

Proof of Theorem sb6

Step	Hyp	Ref	Expression
1		sb1 1810	. . 3 |- ( [ y / x ] ph -> E. x ( x = y /\ ph ) )
2		sb56 2091	. . 3 |- ( E. x ( x = y /\ ph ) <-> A. x ( x = y -> ph ) )
3	1, 2	sylib 201	. 2 |- ( [ y / x ] ph -> A. x ( x = y -> ph ) )
4		sb2 2193	. 2 |- ( A. x ( x = y -> ph ) -> [ y / x ] ph )
5	3, 4	impbii 192	1 |- ( [ y / x ] ph <-> A. x ( x = y -> ph ) )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 375   A.wal 1452   E.wex 1673   [wsb 1807

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-10 1925  ax-12 1943  ax-13 2101

This theorem depends on definitions:  df-bi 190  df-an 377  df-ex 1674  df-nf 1678  df-sb 1808

This theorem is referenced by:  sb5  2269  2sb6  2283  sb6a  2287  2eu6  2397