Theorem sbcopeq1a 4127

Description: Equality theorem for substitution of a class for an ordered pair (analog of sbceq1a 1951, that avoids the existential quantifiers of copsexg 2808).

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 1951	. . 3 |- (y = (2nd` A) -> (ph <-> [(2nd` A) / y]ph))
2		sbceq1a 1951	. . 3 |- (x = (1st` A) -> ([(2nd` A) / y]ph <-> [(1st` A) / x][(2nd` A) / y]ph))
3	1, 2	sylan9bb 543	. 2 |- ((y = (2nd` A) /\ x = (1st` A)) -> (ph <-> [(1st` A) / x][(2nd` A) / y]ph))
4		fveq2 3740	. . 3 |- (<.x, y>. = A -> (2nd` <.x, y>.) = (2nd` A))
5		visset 1820	. . . 4 |- x e. V
6		visset 1820	. . . 4 |- y e. V
7	5, 6	op2nd 4102	. . 3 |- (2nd` <.x, y>.) = y
8	4, 7	syl5eqr 1528	. 2 |- (<.x, y>. = A -> y = (2nd` A))
9		fveq2 3740	. . 3 |- (<.x, y>. = A -> (1st` <.x, y>.) = (1st` A))
10	5	op1st 4101	. . 3 |- (1st` <.x, y>.) = x
11	9, 10	syl5eqr 1528	. 2 |- (<.x, y>. = A -> x = (1st` A))
12	3, 8, 11	sylanc 474	1 |- (<.x, y>. = A -> (ph <-> [(1st` A) / x][(2nd` A) / y]ph))

Syntax hints:   -> wi 3   <-> wb 146   = wceq 960  [wsbc 1174  <.cop 2423  ` cfv 3198  1stc1st 4093  2ndc2nd 4094

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 966  ax-gen 967  ax-8 968  ax-9 969  ax-10 970  ax-11 971  ax-12 972  ax-13 973  ax-14 974  ax-17 975  ax-4 977  ax-5o 979  ax-6o 982  ax-9o 1127  ax-10o 1144  ax-16 1214  ax-11o 1222  ax-ext 1464  ax-sep 2718  ax-nul 2725  ax-pow 2758  ax-pr 2795  ax-un 2882

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 985  df-sb 1176  df-eu 1386  df-mo 1387  df-clab 1470  df-cleq 1475  df-clel 1478  df-ne 1594  df-ral 1656  df-rex 1657  df-v 1819  df-sbc 1949  df-dif 2060  df-un 2061  df-in 2062  df-ss 2064  df-nul 2292  df-pw 2414  df-sn 2424  df-pr 2425  df-op 2428  df-uni 2518  df-br 2635  df-opab 2682  df-id 2851  df-xp 3200  df-rel 3201  df-cnv 3202  df-co 3203  df-dm 3204  df-rn 3205  df-res 3206  df-ima 3207  df-fun 3208  df-fv 3214  df-1st 4095  df-2nd 4096

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