Theorem sbcopeq1a 6085

Description: Equality theorem for substitution of a class for an ordered pair (analog of sbceq1a 2918 that avoids the existential quantifiers of copsexg 4166). (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 2689	. . . . 5 |- x e. _V
2		vex 2689	. . . . 5 |- y e. _V
3	1, 2	op2ndd 6047	. . . 4 |- ( A = <. x , y >. -> ( 2nd ` A ) = y )
4	3	eqcomd 2145	. . 3 |- ( A = <. x , y >. -> y = ( 2nd ` A ) )
5		sbceq1a 2918	. . 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 6046	. . . 4 |- ( A = <. x , y >. -> ( 1st ` A ) = x )
8	7	eqcomd 2145	. . 3 |- ( A = <. x , y >. -> x = ( 1st ` A ) )
9		sbceq1a 2918	. . 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 188	1 |- ( A = <. x , y >. -> ( [. ( 1st ` A ) / x ]. [. ( 2nd ` A ) / y ]. ph <-> ph ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1331   [.wsbc 2909   <.cop 3530   ` cfv 5123   1stc1st 6036   2ndc2nd 6037

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-iota 5088  df-fun 5125  df-fv 5131  df-1st 6038  df-2nd 6039

This theorem is referenced by:  dfopab2  6087  dfoprab3s  6088