Theorem moxfr 37255

Description: Transfer at-most-one between related expressions. (Contributed by Stefan O'Rear, 12-Feb-2015.)

Hypotheses

Ref	Expression
moxfr.a	|- A e. _V
moxfr.b	|- E! y x = A
moxfr.c	|- ( x = A -> ( ph <-> ps ) )

Assertion

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

Distinct variable groups:   ps, x   ph, y   x, A   x, y

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

Proof of Theorem moxfr

Step	Hyp	Ref	Expression
1		moxfr.a	. . . . . 6 |- A e. _V
2	1	a1i 11	. . . . 5 |- ( y e. _V -> A e. _V )
3		moxfr.b	. . . . . . . 8 |- E! y x = A
4		euex 2494	. . . . . . . 8 |- ( E! y x = A -> E. y x = A )
5	3, 4	ax-mp 5	. . . . . . 7 |- E. y x = A
6		rexv 3220	. . . . . . 7 |- ( E. y e. _V x = A <-> E. y x = A )
7	5, 6	mpbir 221	. . . . . 6 |- E. y e. _V x = A
8	7	a1i 11	. . . . 5 |- ( x e. _V -> E. y e. _V x = A )
9		moxfr.c	. . . . 5 |- ( x = A -> ( ph <-> ps ) )
10	2, 8, 9	rexxfr 4888	. . . 4 |- ( E. x e. _V ph <-> E. y e. _V ps )
11		rexv 3220	. . . 4 |- ( E. x e. _V ph <-> E. x ph )
12		rexv 3220	. . . 4 |- ( E. y e. _V ps <-> E. y ps )
13	10, 11, 12	3bitr3i 290	. . 3 |- ( E. x ph <-> E. y ps )
14	1, 3, 9	euxfr 3392	. . 3 |- ( E! x ph <-> E! y ps )
15	13, 14	imbi12i 340	. 2 |- ( ( E. x ph -> E! x ph ) <-> ( E. y ps -> E! y ps ) )
16		df-mo 2475	. 2 |- ( E* x ph <-> ( E. x ph -> E! x ph ) )
17		df-mo 2475	. 2 |- ( E* y ps <-> ( E. y ps -> E! y ps ) )
18	15, 16, 17	3bitr4i 292	1 |- ( E* x ph <-> E* y ps )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   E.wex 1704   e. wcel 1990   E!weu 2470   E*wmo 2471   E.wrex 2913   _Vcvv 3200

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

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  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202

This theorem is referenced by: (None)