Theorem opabbidv 3814

Description: Equivalent wff's yield equal ordered-pair class abstractions (deduction rule). (Contributed by NM, 15-May-1995.)

Hypothesis

Ref	Expression
opabbidv.1	|- (ph -> (ps <-> ch))

Assertion

Ref	Expression
opabbidv	|- (ph -> { <.x , y >. | ps } = { <.x , y >. | ch })

Distinct variable groups:   ph,x   ph,y

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

Proof of Theorem opabbidv

Step	Hyp	Ref	Expression
1		nfv 1418	. 2 |- F/xph
2		nfv 1418	. 2 |- F/yph
3		opabbidv.1	. 2 |- (ph -> (ps <-> ch))
4	1, 2, 3	opabbid 3813	1 |- (ph -> { <.x , y >. | ps } = { <.x , y >. | ch })

Syntax hints:   -> wi 4   <-> wb 98   = wceq 1242   {copab 3808

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-opab 3810

This theorem is referenced by:  opabbii  3815  csbopabg  3826  xpeq1  4302  xpeq2  4303  opabbi2dv  4428  csbcnvg  4462  resopab2  4598  cores  4767  xpcom  4807  dffn5im  5162  f1oiso2  5409  f1ocnvd  5644  ofreq  5657  sprmpt2  5798