Theorem cbvmo 2506

Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 9-Mar-1995.) (Revised by Andrew Salmon, 8-Jun-2011.)

Hypotheses

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

Assertion

Ref	Expression
cbvmo	|- ( E* x ph <-> E* y ps )

Proof of Theorem cbvmo

Step	Hyp	Ref	Expression
1		cbvmo.1	. . . 4 |- F/ y ph
2		cbvmo.2	. . . 4 |- F/ x ps
3		cbvmo.3	. . . 4 |- ( x = y -> ( ph <-> ps ) )
4	1, 2, 3	cbvex 2272	. . 3 |- ( E. x ph <-> E. y ps )
5	1, 2, 3	cbveu 2505	. . 3 |- ( E! x ph <-> E! y ps )
6	4, 5	imbi12i 340	. 2 |- ( ( E. x ph -> E! x ph ) <-> ( E. y ps -> E! y ps ) )
7		df-mo 2475	. 2 |- ( E* x ph <-> ( E. x ph -> E! x ph ) )
8		df-mo 2475	. 2 |- ( E* y ps <-> ( E. y ps -> E! y ps ) )
9	6, 7, 8	3bitr4i 292	1 |- ( E* x ph <-> E* y ps )

Syntax hints:   -> wi 4   <-> wb 196   E.wex 1704   F/wnf 1708   E!weu 2470   E*wmo 2471

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475

This theorem is referenced by:  dffun6f  5902  opabiotafun  6259  2ndcdisj  21259  cbvdisjf  29385  phpreu  33393