Theorem exopxfr2 5266

Description: Transfer ordered-pair existence from/to single variable existence. (Contributed by NM, 26-Feb-2014.)

Hypothesis

Ref	Expression
exopxfr2.1	|- ( x = <. y , z >. -> ( ph <-> ps ) )

Assertion

Ref	Expression
exopxfr2	|- ( Rel A -> ( E. x e. A ph <-> E. y E. z ( <. y , z >. e. A /\ ps ) ) )

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

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

Proof of Theorem exopxfr2

Step	Hyp	Ref	Expression
1		df-rel 5121	. . . . . . 7 |- ( Rel A <-> A C_ ( _V X. _V ) )
2	1	biimpi 206	. . . . . 6 |- ( Rel A -> A C_ ( _V X. _V ) )
3	2	sseld 3602	. . . . 5 |- ( Rel A -> ( x e. A -> x e. ( _V X. _V ) ) )
4	3	adantrd 484	. . . 4 |- ( Rel A -> ( ( x e. A /\ ph ) -> x e. ( _V X. _V ) ) )
5	4	pm4.71rd 667	. . 3 |- ( Rel A -> ( ( x e. A /\ ph ) <-> ( x e. ( _V X. _V ) /\ ( x e. A /\ ph ) ) ) )
6	5	rexbidv2 3048	. 2 |- ( Rel A -> ( E. x e. A ph <-> E. x e. ( _V X. _V ) ( x e. A /\ ph ) ) )
7		eleq1 2689	. . . 4 |- ( x = <. y , z >. -> ( x e. A <-> <. y , z >. e. A ) )
8		exopxfr2.1	. . . 4 |- ( x = <. y , z >. -> ( ph <-> ps ) )
9	7, 8	anbi12d 747	. . 3 |- ( x = <. y , z >. -> ( ( x e. A /\ ph ) <-> ( <. y , z >. e. A /\ ps ) ) )
10	9	exopxfr 5265	. 2 |- ( E. x e. ( _V X. _V ) ( x e. A /\ ph ) <-> E. y E. z ( <. y , z >. e. A /\ ps ) )
11	6, 10	syl6bb 276	1 |- ( Rel A -> ( E. x e. A ph <-> E. y E. z ( <. y , z >. e. A /\ ps ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   E.wrex 2913   _Vcvv 3200   C_ wss 3574   <.cop 4183   X. cxp 5112   Rel wrel 5119

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  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-iun 4522  df-opab 4713  df-xp 5120  df-rel 5121

This theorem is referenced by:  dvhopellsm  36406