Theorem sbcopeq1a 6349

Description: Equality theorem for substitution of a class for an ordered pair (analog of sbceq1a 3041 that avoids the existential quantifiers of copsexg 4336). (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 2805	. . . . 5 |- x e. _V
2		vex 2805	. . . . 5 |- y e. _V
3	1, 2	op2ndd 6311	. . . 4 |- ( A = <. x , y >. -> ( 2nd ` A ) = y )
4	3	eqcomd 2237	. . 3 |- ( A = <. x , y >. -> y = ( 2nd ` A ) )
5		sbceq1a 3041	. . 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 6310	. . . 4 |- ( A = <. x , y >. -> ( 1st ` A ) = x )
8	7	eqcomd 2237	. . 3 |- ( A = <. x , y >. -> x = ( 1st ` A ) )
9		sbceq1a 3041	. . 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 1397   [.wsbc 3031   <.cop 3672   ` cfv 5326   1stc1st 6300   2ndc2nd 6301

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-iota 5286  df-fun 5328  df-fv 5334  df-1st 6302  df-2nd 6303

This theorem is referenced by:  dfopab2  6351  dfoprab3s  6352