Theorem sbcopeq1a 6202

Description: Equality theorem for substitution of a class for an ordered pair (analog of sbceq1a 2984 that avoids the existential quantifiers of copsexg 4256). (Contributed by NM, 19-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)

Assertion

Ref	Expression
sbcopeq1a	|- ( A = <. x , y >. -> ( [. ( 1st ` A ) / x ]. [. ( 2nd ` A ) / y ]. ph <-> ph ) )

Proof of Theorem sbcopeq1a

Step	Hyp	Ref	Expression
1		vex 2752	. . . . 5 |- x e. _V
2		vex 2752	. . . . 5 |- y e. _V
3	1, 2	op2ndd 6164	. . . 4 |- ( A = <. x , y >. -> ( 2nd ` A ) = y )
4	3	eqcomd 2193	. . 3 |- ( A = <. x , y >. -> y = ( 2nd ` A ) )
5		sbceq1a 2984	. . 3 |- ( y = ( 2nd ` A ) -> ( ph <-> [. ( 2nd ` A ) / y ]. ph ) )
6	4, 5	syl 14	. 2 |- ( A = <. x , y >. -> ( ph <-> [. ( 2nd ` A ) / y ]. ph ) )
7	1, 2	op1std 6163	. . . 4 |- ( A = <. x , y >. -> ( 1st ` A ) = x )
8	7	eqcomd 2193	. . 3 |- ( A = <. x , y >. -> x = ( 1st ` A ) )
9		sbceq1a 2984	. . 3 |- ( x = ( 1st ` A ) -> ( [. ( 2nd ` A ) / y ]. ph <-> [. ( 1st ` A ) / x ]. [. ( 2nd ` A ) / y ]. ph ) )
10	8, 9	syl 14	. 2 |- ( A = <. x , y >. -> ( [. ( 2nd ` A ) / y ]. ph <-> [. ( 1st ` A ) / x ]. [. ( 2nd ` A ) / y ]. ph ) )
11	6, 10	bitr2d 189	1 |- ( A = <. x , y >. -> ( [. ( 1st ` A ) / x ]. [. ( 2nd ` A ) / y ]. ph <-> ph ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1363   [.wsbc 2974   <.cop 3607   ` cfv 5228   1stc1st 6153   2ndc2nd 6154

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-sbc 2975  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-iota 5190  df-fun 5230  df-fv 5236  df-1st 6155  df-2nd 6156

This theorem is referenced by:  dfopab2  6204  dfoprab3s  6205