Theorem equsex 1962

Description: A useful equivalence related to substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.)

Hypotheses

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

Assertion

Ref	Expression
equsex	|- (E.x(x = y /\ ph) <-> ps)

Proof of Theorem equsex

Step	Hyp	Ref	Expression
1		exnal 1574	. 2 |- (E.x -. (x = y -> -. ph) <-> -. A.x(x = y -> -. ph))
2		df-an 360	. . 3 |- ((x = y /\ ph) <-> -. (x = y -> -. ph))
3	2	exbii 1582	. 2 |- (E.x(x = y /\ ph) <-> E.x -. (x = y -> -. ph))
4		equsex.1	. . . . 5 |- F/xps
5	4	nfn 1793	. . . 4 |- F/x -. ps
6		equsex.2	. . . . 5 |- (x = y -> (ph <-> ps))
7	6	notbid 285	. . . 4 |- (x = y -> (-. ph <-> -. ps))
8	5, 7	equsal 1960	. . 3 |- (A.x(x = y -> -. ph) <-> -. ps)
9	8	con2bii 322	. 2 |- (ps <-> -. A.x(x = y -> -. ph))
10	1, 3, 9	3bitr4i 268	1 |- (E.x(x = y /\ ph) <-> ps)

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176   /\ wa 358  A.wal 1540  E.wex 1541   F/wnf 1544

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545

This theorem is referenced by:  equsexh  1963  cleljustALT  2015  sb56  2098  sb10f  2122