Theorem cbvrex 2833

Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 31-Jul-2003.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)

Hypotheses

Ref	Expression
cbvral.1	|- F/yph
cbvral.2	|- F/xps
cbvral.3	|- (x = y -> (ph <-> ps))

Assertion

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

Distinct variable groups:   x,A   y,A

Allowed substitution hints:   ph(x,y)   ps(x,y)

Proof of Theorem cbvrex

Step	Hyp	Ref	Expression
1		nfcv 2490	. 2 |- F/_xA
2		nfcv 2490	. 2 |- F/_yA
3		cbvral.1	. 2 |- F/yph
4		cbvral.2	. 2 |- F/xps
5		cbvral.3	. 2 |- (x = y -> (ph <-> ps))
6	1, 2, 3, 4, 5	cbvrexf 2831	1 |- (E.x e. A ph <-> E.y e. A ps)

Syntax hints:   -> wi 4   <-> wb 176   F/wnf 1544  E.wrex 2616

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 2479  df-ral 2620  df-rex 2621

This theorem is referenced by:  cbvrmo  2835  cbvrexv  2837  cbvrexsv  2848  cbviun  4004