Theorem reusv3i 4875

Description: Two ways of expressing existential uniqueness via an indirect equality. (Contributed by NM, 23-Dec-2012.)

Hypotheses

Ref	Expression
reusv3.1	|- ( y = z -> ( ph <-> ps ) )
reusv3.2	|- ( y = z -> C = D )

Assertion

Ref	Expression
reusv3i	|- ( E. x e. A A. y e. B ( ph -> x = C ) -> A. y e. B A. z e. B ( ( ph /\ ps ) -> C = D ) )

Distinct variable groups:   x, y, z, B   x, C, z   x, D, y   ph, x, z   ps, x, y

Allowed substitution hints:   ph( y)   ps( z)   A( x, y, z)   C( y)   D( z)

Proof of Theorem reusv3i

Step	Hyp	Ref	Expression
1		reusv3.1	. . . . . 6 |- ( y = z -> ( ph <-> ps ) )
2		reusv3.2	. . . . . . 7 |- ( y = z -> C = D )
3	2	eqeq2d 2632	. . . . . 6 |- ( y = z -> ( x = C <-> x = D ) )
4	1, 3	imbi12d 334	. . . . 5 |- ( y = z -> ( ( ph -> x = C ) <-> ( ps -> x = D ) ) )
5	4	cbvralv 3171	. . . 4 |- ( A. y e. B ( ph -> x = C ) <-> A. z e. B ( ps -> x = D ) )
6	5	biimpi 206	. . 3 |- ( A. y e. B ( ph -> x = C ) -> A. z e. B ( ps -> x = D ) )
7		raaanv 4083	. . . 4 |- ( A. y e. B A. z e. B ( ( ph -> x = C ) /\ ( ps -> x = D ) ) <-> ( A. y e. B ( ph -> x = C ) /\ A. z e. B ( ps -> x = D ) ) )
8		prth 595	. . . . . 6 |- ( ( ( ph -> x = C ) /\ ( ps -> x = D ) ) -> ( ( ph /\ ps ) -> ( x = C /\ x = D ) ) )
9		eqtr2 2642	. . . . . 6 |- ( ( x = C /\ x = D ) -> C = D )
10	8, 9	syl6 35	. . . . 5 |- ( ( ( ph -> x = C ) /\ ( ps -> x = D ) ) -> ( ( ph /\ ps ) -> C = D ) )
11	10	2ralimi 2953	. . . 4 |- ( A. y e. B A. z e. B ( ( ph -> x = C ) /\ ( ps -> x = D ) ) -> A. y e. B A. z e. B ( ( ph /\ ps ) -> C = D ) )
12	7, 11	sylbir 225	. . 3 |- ( ( A. y e. B ( ph -> x = C ) /\ A. z e. B ( ps -> x = D ) ) -> A. y e. B A. z e. B ( ( ph /\ ps ) -> C = D ) )
13	6, 12	mpdan 702	. 2 |- ( A. y e. B ( ph -> x = C ) -> A. y e. B A. z e. B ( ( ph /\ ps ) -> C = D ) )
14	13	rexlimivw 3029	1 |- ( E. x e. A A. y e. B ( ph -> x = C ) -> A. y e. B A. z e. B ( ( ph /\ ps ) -> C = D ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   A.wral 2912   E.wrex 2913

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-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-v 3202  df-dif 3577  df-nul 3916

This theorem is referenced by:  reusv3  4876