csbopeq1a - Intuitionistic Logic Explorer

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Theorem csbopeq1a 5756

Description: Equality theorem for substitution of a class A for an ordered pair ⟨x, y⟩ in B (analog of csbeq1a 2854). (Contributed by NM, 19-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)

**Assertion**
Ref	Expression
csbopeq1a	⊢ (A = ⟨x, y⟩ → ⦋(1^st ‘A) / x⦌⦋(2^nd ‘A) / y⦌B = B)

**Proof of Theorem csbopeq1a**
Step	Hyp	Ref	Expression
1		vex 2554	. . . . 5 ⊢ x ∈ V
2		vex 2554	. . . . 5 ⊢ y ∈ V
3	1, 2	op2ndd 5718	. . . 4 ⊢ (A = ⟨x, y⟩ → (2^nd ‘A) = y)
4	3	eqcomd 2042	. . 3 ⊢ (A = ⟨x, y⟩ → y = (2^nd ‘A))
5		csbeq1a 2854	. . 3 ⊢ (y = (2^nd ‘A) → B = ⦋(2^nd ‘A) / y⦌B)
6	4, 5	syl 14	. 2 ⊢ (A = ⟨x, y⟩ → B = ⦋(2^nd ‘A) / y⦌B)
7	1, 2	op1std 5717	. . . 4 ⊢ (A = ⟨x, y⟩ → (1^st ‘A) = x)
8	7	eqcomd 2042	. . 3 ⊢ (A = ⟨x, y⟩ → x = (1^st ‘A))
9		csbeq1a 2854	. . 3 ⊢ (x = (1^st ‘A) → ⦋(2^nd ‘A) / y⦌B = ⦋(1^st ‘A) / x⦌⦋(2^nd ‘A) / y⦌B)
10	8, 9	syl 14	. 2 ⊢ (A = ⟨x, y⟩ → ⦋(2^nd ‘A) / y⦌B = ⦋(1^st ‘A) / x⦌⦋(2^nd ‘A) / y⦌B)
11	6, 10	eqtr2d 2070	1 ⊢ (A = ⟨x, y⟩ → ⦋(1^st ‘A) / x⦌⦋(2^nd ‘A) / y⦌B = B)

Colors of variables: wff set class

Syntax hints: → wi 4 = wceq 1242 ⦋csb 2846 ⟨cop 3370 ‘cfv 4845 1^st c1st 5707 2^nd c2nd 5708

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-13 1401 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 ax-un 4136

This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-v 2553 df-sbc 2759 df-csb 2847 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-br 3756 df-opab 3810 df-mpt 3811 df-id 4021 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-rn 4299 df-iota 4810 df-fun 4847 df-fv 4853 df-1st 5709 df-2nd 5710

This theorem is referenced by: dfmpt2 5786

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