Theorem opabbid 4625

Description: Equivalent wff's yield equal ordered-pair class abstractions (deduction rule). (Contributed by NM, 21-Feb-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Hypotheses

Ref	Expression
opabbid.1	|- F/xph
opabbid.2	|- F/yph
opabbid.3	|- (ph -> (ps <-> ch))

Assertion

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

Proof of Theorem opabbid

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		opabbid.1	. . . 4 |- F/xph
2		opabbid.2	. . . . 5 |- F/yph
3		opabbid.3	. . . . . 6 |- (ph -> (ps <-> ch))
4	3	anbi2d 684	. . . . 5 |- (ph -> ((z = <.x, y>. /\ ps) <-> (z = <.x, y>. /\ ch)))
5	2, 4	exbid 1773	. . . 4 |- (ph -> (E.y(z = <.x, y>. /\ ps) <-> E.y(z = <.x, y>. /\ ch)))
6	1, 5	exbid 1773	. . 3 |- (ph -> (E.xE.y(z = <.x, y>. /\ ps) <-> E.xE.y(z = <.x, y>. /\ ch)))
7	6	abbidv 2468	. 2 |- (ph -> {z | E.xE.y(z = <.x, y>. /\ ps)} = {z | E.xE.y(z = <.x, y>. /\ ch)})
8		df-opab 4624	. 2 |- {<.x, y>. | ps} = {z | E.xE.y(z = <.x, y>. /\ ps)}
9		df-opab 4624	. 2 |- {<.x, y>. | ch} = {z | E.xE.y(z = <.x, y>. /\ ch)}
10	7, 8, 9	3eqtr4g 2410	1 |- (ph -> {<.x, y>. | ps} = {<.x, y>. | ch})

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  E.wex 1541   F/wnf 1544   = wceq 1642  {cab 2339  <.cop 4562  {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:  opabbidv  4626  fnoprabg  5586  mpteq12f  5656