Theorem cbveu 2224

Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 25-Nov-1994.) (Revised by Mario Carneiro, 7-Oct-2016.)

Hypotheses

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

Assertion

Ref	Expression
cbveu	|- (E!xph <-> E!yps)

Proof of Theorem cbveu

Step	Hyp	Ref	Expression
1		cbveu.1	. . 3 |- F/yph
2	1	sb8eu 2222	. 2 |- (E!xph <-> E!y[y / x]ph)
3		cbveu.2	. . . 4 |- F/xps
4		cbveu.3	. . . 4 |- (x = y -> (ph <-> ps))
5	3, 4	sbie 2038	. . 3 |- ([y / x]ph <-> ps)
6	5	eubii 2213	. 2 |- (E!y[y / x]ph <-> E!yps)
7	2, 6	bitri 240	1 |- (E!xph <-> E!yps)

Syntax hints:   -> wi 4   <-> wb 176   F/wnf 1544  [wsb 1648  E!weu 2204

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-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208

This theorem is referenced by:  cbvmo  2241  cbvreu  2834  cbvreucsf  3201  tz6.12-1  5345