Theorem cbveu 2078

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/ y ph
cbveu.2	|- F/ x ps
cbveu.3	|- ( x = y -> ( ph <-> ps ) )

Assertion

Ref	Expression
cbveu	|- ( E! x ph <-> E! y ps )

Proof of Theorem cbveu

Step	Hyp	Ref	Expression
1		cbveu.1	. . 3 |- F/ y ph
2	1	sb8eu 2067	. 2 |- ( E! x ph <-> E! y [ y / x ] ph )
3		cbveu.2	. . . 4 |- F/ x ps
4		cbveu.3	. . . 4 |- ( x = y -> ( ph <-> ps ) )
5	3, 4	sbie 1814	. . 3 |- ( [ y / x ] ph <-> ps )
6	5	eubii 2063	. 2 |- ( E! y [ y / x ] ph <-> E! y ps )
7	2, 6	bitri 184	1 |- ( E! x ph <-> E! y ps )

Syntax hints:   -> wi 4   <-> wb 105   F/wnf 1483   [wsb 1785   E!weu 2054

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057

This theorem is referenced by:  cbvmo  2094  cbvreu  2736  cbvreucsf  3158  tz6.12f  5605  f1ompt  5731  climeu  11607