Theorem opabbidv 4626

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 1619	. 2 |- F/xph
2		nfv 1619	. 2 |- F/yph
3		opabbidv.1	. 2 |- (ph -> (ps <-> ch))
4	1, 2, 3	opabbid 4625	1 |- (ph -> {<.x, y>. | ps} = {<.x, y>. | ch})

Syntax hints:   -> wi 4   <-> wb 176   = wceq 1642  {copab 4623

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-opab 4624

This theorem is referenced by:  opabbii  4627  csbopabg  4638  xpeq1  4799  xpeq2  4800  opabbi2dv  4868  resopab2  5002  cores  5085  f1oiso2  5501  f1od  5727