Theorem dropab2 38652

Description: Theorem to aid use of the distinctor reduction theorem with ordered pair class abstraction. (Contributed by Andrew Salmon, 25-Jul-2011.)

Assertion

Ref	Expression
dropab2	|- ( A. x x = y -> { <. z , x >. | ph } = { <. z , y >. | ph } )

Proof of Theorem dropab2

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		opeq2 4403	. . . . . . . 8 |- ( x = y -> <. z , x >. = <. z , y >. )
2	1	sps 2055	. . . . . . 7 |- ( A. x x = y -> <. z , x >. = <. z , y >. )
3	2	eqeq2d 2632	. . . . . 6 |- ( A. x x = y -> ( w = <. z , x >. <-> w = <. z , y >. ) )
4	3	anbi1d 741	. . . . 5 |- ( A. x x = y -> ( ( w = <. z , x >. /\ ph ) <-> ( w = <. z , y >. /\ ph ) ) )
5	4	drex1 2327	. . . 4 |- ( A. x x = y -> ( E. x ( w = <. z , x >. /\ ph ) <-> E. y ( w = <. z , y >. /\ ph ) ) )
6	5	drex2 2328	. . 3 |- ( A. x x = y -> ( E. z E. x ( w = <. z , x >. /\ ph ) <-> E. z E. y ( w = <. z , y >. /\ ph ) ) )
7	6	abbidv 2741	. 2 |- ( A. x x = y -> { w | E. z E. x ( w = <. z , x >. /\ ph ) } = { w | E. z E. y ( w = <. z , y >. /\ ph ) } )
8		df-opab 4713	. 2 |- { <. z , x >. | ph } = { w | E. z E. x ( w = <. z , x >. /\ ph ) }
9		df-opab 4713	. 2 |- { <. z , y >. | ph } = { w | E. z E. y ( w = <. z , y >. /\ ph ) }
10	7, 8, 9	3eqtr4g 2681	1 |- ( A. x x = y -> { <. z , x >. | ph } = { <. z , y >. | ph } )

Syntax hints:   -> wi 4   /\ wa 384   A.wal 1481   = wceq 1483   E.wex 1704   {cab 2608   <.cop 4183   {copab 4712

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

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-rab 2921  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

This theorem is referenced by: (None)