Theorem copsex2gb 5230

Description: Implicit substitution inference for ordered pairs. Compare copsex2ga 5231. (Contributed by NM, 12-Mar-2014.)

Hypothesis

Ref	Expression
copsex2ga.1	|- ( A = <. x , y >. -> ( ph <-> ps ) )

Assertion

Ref	Expression
copsex2gb	|- ( E. x E. y ( A = <. x , y >. /\ ps ) <-> ( A e. ( _V X. _V ) /\ ph ) )

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

Allowed substitution hints:   ps( x, y)

Proof of Theorem copsex2gb

Step	Hyp	Ref	Expression
1		elvv 5177	. . 3 |- ( A e. ( _V X. _V ) <-> E. x E. y A = <. x , y >. )
2	1	anbi1i 731	. 2 |- ( ( A e. ( _V X. _V ) /\ ph ) <-> ( E. x E. y A = <. x , y >. /\ ph ) )
3		19.41vv 1915	. 2 |- ( E. x E. y ( A = <. x , y >. /\ ph ) <-> ( E. x E. y A = <. x , y >. /\ ph ) )
4		copsex2ga.1	. . . 4 |- ( A = <. x , y >. -> ( ph <-> ps ) )
5	4	pm5.32i 669	. . 3 |- ( ( A = <. x , y >. /\ ph ) <-> ( A = <. x , y >. /\ ps ) )
6	5	2exbii 1775	. 2 |- ( E. x E. y ( A = <. x , y >. /\ ph ) <-> E. x E. y ( A = <. x , y >. /\ ps ) )
7	2, 3, 6	3bitr2ri 289	1 |- ( E. x E. y ( A = <. x , y >. /\ ps ) <-> ( A e. ( _V X. _V ) /\ ph ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   _Vcvv 3200   <.cop 4183   X. cxp 5112

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-v 3202  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-opab 4713  df-xp 5120

This theorem is referenced by:  copsex2ga  5231  elopaba  5232  elcnvlem  37907