Theorem cbvopabv 2673

Description: Rule used to change bound variables in an ordered-pair class abstraction, using implicit substitution.

Hypothesis

Ref	Expression
cbvopabv.1	|- ((x = z /\ y = w) -> (ph <-> ps))

Assertion

Ref	Expression
cbvopabv	|- {<.x, y>. | ph} = {<.z, w>. | ps}

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

Proof of Theorem cbvopabv

Step	Hyp	Ref	Expression
1		ax-17 971	. 2 |- (ph -> A.zph)
2		ax-17 971	. 2 |- (ph -> A.wph)
3		ax-17 971	. 2 |- (ps -> A.xps)
4		ax-17 971	. 2 |- (ps -> A.yps)
5		cbvopabv.1	. 2 |- ((x = z /\ y = w) -> (ph <-> ps))
6	1, 2, 3, 4, 5	cbvopab 2672	1 |- {<.x, y>. | ph} = {<.z, w>. | ps}

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956  {copab 2666

This theorem is referenced by:  inf3lema 4609  numth 4784  zorn2 4796  seq1suclem 6316  fsumcnlem 7989  effoi 8745  cmpbva 10464  trnij 10637  cnvtr 10638

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-un 2050  df-sn 2412  df-pr 2413  df-op 2416  df-opab 2667

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