Theorem sbcopeq1a 4117

Description: Equality theorem for substitution of a class for an ordered pair (analog of sbceq1a 1947, that avoids the existential quantifiers of copsexg 2798).

Assertion

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

Proof of Theorem sbcopeq1a

Step	Hyp	Ref	Expression
1		sbceq1a 1947	. . 3 |- (y = (2nd` A) -> (ph <-> [(2nd` A) / y]ph))
2		sbceq1a 1947	. . 3 |- (x = (1st` A) -> ([(2nd` A) / y]ph <-> [(1st` A) / x][(2nd` A) / y]ph))
3	1, 2	sylan9bb 542	. 2 |- ((y = (2nd` A) /\ x = (1st` A)) -> (ph <-> [(1st` A) / x][(2nd` A) / y]ph))
4		fveq2 3730	. . 3 |- (<.x, y>. = A -> (2nd` <.x, y>.) = (2nd` A))
5		visset 1816	. . . 4 |- x e. V
6		visset 1816	. . . 4 |- y e. V
7	5, 6	op2nd 4092	. . 3 |- (2nd` <.x, y>.) = y
8	4, 7	syl5eqr 1524	. 2 |- (<.x, y>. = A -> y = (2nd` A))
9		fveq2 3730	. . 3 |- (<.x, y>. = A -> (1st` <.x, y>.) = (1st` A))
10	5	op1st 4091	. . 3 |- (1st` <.x, y>.) = x
11	9, 10	syl5eqr 1524	. 2 |- (<.x, y>. = A -> x = (1st` A))
12	3, 8, 11	sylanc 473	1 |- (<.x, y>. = A -> (ph <-> [(1st` A) / x][(2nd` A) / y]ph))

Syntax hints:   -> wi 3   <-> wb 146   = wceq 958  [wsbc 1172  <.cop 2415  ` cfv 3188  1stc1st 4083  2ndc2nd 4084

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-sbc 1945  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fv 3204  df-1st 4085  df-2nd 4086

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