Theorem cbvrexf 2830

Description: Rule used to change bound variables, using implicit substitution. (Contributed by FL, 27-Apr-2008.) (Revised by Mario Carneiro, 9-Oct-2016.)

Hypotheses

Ref	Expression
cbvralf.1	|- F/_xA
cbvralf.2	|- F/_yA
cbvralf.3	|- F/yph
cbvralf.4	|- F/xps
cbvralf.5	|- (x = y -> (ph <-> ps))

Assertion

Ref	Expression
cbvrexf	|- (E.x e. A ph <-> E.y e. A ps)

Proof of Theorem cbvrexf

Step	Hyp	Ref	Expression
1		cbvralf.1	. . . 4 |- F/_xA
2		cbvralf.2	. . . 4 |- F/_yA
3		cbvralf.3	. . . . 5 |- F/yph
4	3	nfn 1793	. . . 4 |- F/y -. ph
5		cbvralf.4	. . . . 5 |- F/xps
6	5	nfn 1793	. . . 4 |- F/x -. ps
7		cbvralf.5	. . . . 5 |- (x = y -> (ph <-> ps))
8	7	notbid 285	. . . 4 |- (x = y -> (-. ph <-> -. ps))
9	1, 2, 4, 6, 8	cbvralf 2829	. . 3 |- (A.x e. A -. ph <-> A.y e. A -. ps)
10	9	notbii 287	. 2 |- (-. A.x e. A -. ph <-> -. A.y e. A -. ps)
11		dfrex2 2627	. 2 |- (E.x e. A ph <-> -. A.x e. A -. ph)
12		dfrex2 2627	. 2 |- (E.y e. A ps <-> -. A.y e. A -. ps)
13	10, 11, 12	3bitr4i 268	1 |- (E.x e. A ph <-> E.y e. A ps)

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176   F/wnf 1544   = wceq 1642   F/_wnfc 2476  A.wral 2614  E.wrex 2615

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  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-rex 2620

This theorem is referenced by:  cbvrex  2832