Theorem csbopeq1a 4112

Description: Equality theorem for substitution of a class A for an ordered pair <.x, y>. in B (analog of csbeq1a 2006).

Assertion

Ref	Expression
csbopeq1a	|- (<.x, y>. = A -> B = [_(1st` A) / x]_[_(2nd` A) / y]_B)

Proof of Theorem csbopeq1a

Step	Hyp	Ref	Expression
1		csbeq1a 2006	. . 3 |- (y = (2nd` A) -> B = [_(2nd` A) / y]_B)
2		csbeq1a 2006	. . 3 |- (x = (1st` A) -> [_(2nd` A) / y]_B = [_(1st` A) / x]_[_(2nd` A) / y]_B)
3	1, 2	sylan9eq 1527	. 2 |- ((y = (2nd` A) /\ x = (1st` A)) -> B = [_(1st` A) / x]_[_(2nd` A) / y]_B)
4		fveq2 3724	. . 3 |- (<.x, y>. = A -> (2nd` <.x, y>.) = (2nd` A))
5		visset 1813	. . . 4 |- x e. V
6		visset 1813	. . . 4 |- y e. V
7	5, 6	op2nd 4086	. . 3 |- (2nd` <.x, y>.) = y
8	4, 7	syl5eqr 1521	. 2 |- (<.x, y>. = A -> y = (2nd` A))
9		fveq2 3724	. . 3 |- (<.x, y>. = A -> (1st` <.x, y>.) = (1st` A))
10	5	op1st 4085	. . 3 |- (1st` <.x, y>.) = x
11	9, 10	syl5eqr 1521	. 2 |- (<.x, y>. = A -> x = (1st` A))
12	3, 8, 11	sylanc 471	1 |- (<.x, y>. = A -> B = [_(1st` A) / x]_[_(2nd` A) / y]_B)

Syntax hints:   -> wi 3   = wceq 956  [_csb 2001  <.cop 2411  ` cfv 3182  1stc1st 4077  2ndc2nd 4078

This theorem is referenced by:  foprab2 4119

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fv 3198  df-1st 4079  df-2nd 4080

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