Theorem sbcopeq1a 4095

Description: Equality theorem for substitution of a class for an ordered pair (analog of sbceq1a 1934, that avoids the existential quantifiers of copsexg 2782).

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 1934	. . 3 |- (y = (2nd` A) -> (ph <-> [(2nd` A) / y]ph))
2		sbceq1a 1934	. . 3 |- (x = (1st` A) -> ([(2nd` A) / y]ph <-> [(1st` A) / x][(2nd` A) / y]ph))
3	1, 2	sylan9bb 538	. 2 |- ((y = (2nd` A) /\ x = (1st` A)) -> (ph <-> [(1st` A) / x][(2nd` A) / y]ph))
4		fveq2 3709	. . 3 |- (<.x, y>. = A -> (2nd` <.x, y>.) = (2nd` A))
5		visset 1804	. . . 4 |- x e. V
6		visset 1804	. . . 4 |- y e. V
7	5, 6	op2nd 4070	. . 3 |- (2nd` <.x, y>.) = y
8	4, 7	syl5eqr 1513	. 2 |- (<.x, y>. = A -> y = (2nd` A))
9		fveq2 3709	. . 3 |- (<.x, y>. = A -> (1st` <.x, y>.) = (1st` A))
10	5	op1st 4069	. . 3 |- (1st` <.x, y>.) = x
11	9, 10	syl5eqr 1513	. 2 |- (<.x, y>. = A -> x = (1st` A))
12	3, 8, 11	sylanc 471	1 |- (<.x, y>. = A -> (ph <-> [(1st` A) / x][(2nd` A) / y]ph))

Syntax hints:   -> wi 3   <-> wb 146   = wceq 953  [wsbc 1166  <.cop 2401  ` cfv 3172  1stc1st 4061  2ndc2nd 4062

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-v 1803  df-sbc 1932  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fv 3188  df-1st 4063  df-2nd 4064

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