Theorem cbveu 1921

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 1910	. 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 1671	. . 3 |- ([y / x]ph <-> ps)
6	5	eubii 1906	. 2 |- (E!y[y / x]ph <-> E!yps)
7	2, 6	bitri 173	1 |- (E!xph <-> E!yps)

Syntax hints:   -> wi 4   <-> wb 98   F/wnf 1346  [wsb 1642  E!weu 1897

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900

This theorem is referenced by:  cbvmo  1937  cbvreu  2525  cbvreucsf  2904  tz6.12f  5145  f1ompt  5263