Theorem equsal 2291

Description: An equivalence related to implicit substitution. See equsalvw 1931 and equsalv 2108 for versions with dv conditions proved from fewer axioms. See also the dual form equsex 2292. (Contributed by NM, 2-Jun-1993.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 5-Feb-2018.)

Hypotheses

Ref	Expression
equsal.1	|- F/ x ps
equsal.2	|- ( x = y -> ( ph <-> ps ) )

Assertion

Ref	Expression
equsal	|- ( A. x ( x = y -> ph ) <-> ps )

Proof of Theorem equsal

Step	Hyp	Ref	Expression
1		equsal.1	. . 3 |- F/ x ps
2	1	19.23 2080	. 2 |- ( A. x ( x = y -> ps ) <-> ( E. x x = y -> ps ) )
3		equsal.2	. . . 4 |- ( x = y -> ( ph <-> ps ) )
4	3	pm5.74i 260	. . 3 |- ( ( x = y -> ph ) <-> ( x = y -> ps ) )
5	4	albii 1747	. 2 |- ( A. x ( x = y -> ph ) <-> A. x ( x = y -> ps ) )
6		ax6e 2250	. . 3 |- E. x x = y
7	6	a1bi 352	. 2 |- ( ps <-> ( E. x x = y -> ps ) )
8	2, 5, 7	3bitr4i 292	1 |- ( A. x ( x = y -> ph ) <-> ps )

Syntax hints:   -> wi 4   <-> wb 196   A.wal 1481   E.wex 1704   F/wnf 1708

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-nf 1710

This theorem is referenced by:  equsex  2292  equsalh  2294  dvelimf  2334  sb6x  2384  sb6rf  2423