Theorem rmoxfrd 29333

Description: Transfer "at most one" restricted quantification from a variable x to another variable y contained in expression A. (Contributed by Thierry Arnoux, 7-Apr-2017.) (Revised by Thierry Arnoux, 8-Oct-2017.)

Hypotheses

Ref	Expression
rmoxfrd.1	|- ( ( ph /\ y e. C ) -> A e. B )
rmoxfrd.2	|- ( ( ph /\ x e. B ) -> E! y e. C x = A )
rmoxfrd.3	|- ( ( ph /\ x = A ) -> ( ps <-> ch ) )

Assertion

Ref	Expression
rmoxfrd	|- ( ph -> ( E* x e. B ps <-> E* y e. C ch ) )

Distinct variable groups:   x, A   x, y, B   x, C, y   ph, x, y   ps, y   ch, x

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

Proof of Theorem rmoxfrd

Step	Hyp	Ref	Expression
1		rmoxfrd.1	. . . . . 6 |- ( ( ph /\ y e. C ) -> A e. B )
2		rmoxfrd.2	. . . . . . 7 |- ( ( ph /\ x e. B ) -> E! y e. C x = A )
3		reurex 3160	. . . . . . 7 |- ( E! y e. C x = A -> E. y e. C x = A )
4	2, 3	syl 17	. . . . . 6 |- ( ( ph /\ x e. B ) -> E. y e. C x = A )
5		rmoxfrd.3	. . . . . 6 |- ( ( ph /\ x = A ) -> ( ps <-> ch ) )
6	1, 4, 5	rexxfrd 4881	. . . . 5 |- ( ph -> ( E. x e. B ps <-> E. y e. C ch ) )
7		df-rex 2918	. . . . 5 |- ( E. x e. B ps <-> E. x ( x e. B /\ ps ) )
8		df-rex 2918	. . . . 5 |- ( E. y e. C ch <-> E. y ( y e. C /\ ch ) )
9	6, 7, 8	3bitr3g 302	. . . 4 |- ( ph -> ( E. x ( x e. B /\ ps ) <-> E. y ( y e. C /\ ch ) ) )
10	1, 2, 5	reuxfr4d 29330	. . . . 5 |- ( ph -> ( E! x e. B ps <-> E! y e. C ch ) )
11		df-reu 2919	. . . . 5 |- ( E! x e. B ps <-> E! x ( x e. B /\ ps ) )
12		df-reu 2919	. . . . 5 |- ( E! y e. C ch <-> E! y ( y e. C /\ ch ) )
13	10, 11, 12	3bitr3g 302	. . . 4 |- ( ph -> ( E! x ( x e. B /\ ps ) <-> E! y ( y e. C /\ ch ) ) )
14	9, 13	imbi12d 334	. . 3 |- ( ph -> ( ( E. x ( x e. B /\ ps ) -> E! x ( x e. B /\ ps ) ) <-> ( E. y ( y e. C /\ ch ) -> E! y ( y e. C /\ ch ) ) ) )
15		df-mo 2475	. . 3 |- ( E* x ( x e. B /\ ps ) <-> ( E. x ( x e. B /\ ps ) -> E! x ( x e. B /\ ps ) ) )
16		df-mo 2475	. . 3 |- ( E* y ( y e. C /\ ch ) <-> ( E. y ( y e. C /\ ch ) -> E! y ( y e. C /\ ch ) ) )
17	14, 15, 16	3bitr4g 303	. 2 |- ( ph -> ( E* x ( x e. B /\ ps ) <-> E* y ( y e. C /\ ch ) ) )
18		df-rmo 2920	. 2 |- ( E* x e. B ps <-> E* x ( x e. B /\ ps ) )
19		df-rmo 2920	. 2 |- ( E* y e. C ch <-> E* y ( y e. C /\ ch ) )
20	17, 18, 19	3bitr4g 303	1 |- ( ph -> ( E* x e. B ps <-> E* y e. C ch ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   E!weu 2470   E*wmo 2471   E.wrex 2913   E!wreu 2914   E*wrmo 2915

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-reu 2919  df-rmo 2920  df-v 3202

This theorem is referenced by:  disjrdx  29404