Theorem sbcoteq1a 8056

Description: Equality theorem for substitution of a class for an ordered triple. (Contributed by Scott Fenton, 22-Aug-2024.)

Assertion

Ref	Expression
sbcoteq1a	⊢ (𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ → ([(1st ‘(1st ‘𝐴)) / 𝑥][(2nd ‘(1st ‘𝐴)) / 𝑦][(2nd ‘𝐴) / 𝑧]𝜑 ↔ 𝜑))

Proof of Theorem sbcoteq1a

Step	Hyp	Ref	Expression
1		fveq2 6896	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ → (2nd ‘𝐴) = (2nd ‘⟨𝑥, 𝑦, 𝑧⟩))
2		ot3rdg 8010	. . . . 5 ⊢ (𝑧 ∈ V → (2nd ‘⟨𝑥, 𝑦, 𝑧⟩) = 𝑧)
3	2	elv 3467	. . . 4 ⊢ (2nd ‘⟨𝑥, 𝑦, 𝑧⟩) = 𝑧
4	1, 3	eqtr2di 2782	. . 3 ⊢ (𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ → 𝑧 = (2nd ‘𝐴))
5		sbceq1a 3784	. . 3 ⊢ (𝑧 = (2nd ‘𝐴) → (𝜑 ↔ [(2nd ‘𝐴) / 𝑧]𝜑))
6	4, 5	syl 17	. 2 ⊢ (𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ → (𝜑 ↔ [(2nd ‘𝐴) / 𝑧]𝜑))
7		2fveq3 6901	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ → (2nd ‘(1st ‘𝐴)) = (2nd ‘(1st ‘⟨𝑥, 𝑦, 𝑧⟩)))
8		vex 3465	. . . . 5 ⊢ 𝑥 ∈ V
9		vex 3465	. . . . 5 ⊢ 𝑦 ∈ V
10		vex 3465	. . . . 5 ⊢ 𝑧 ∈ V
11		ot2ndg 8009	. . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (2nd ‘(1st ‘⟨𝑥, 𝑦, 𝑧⟩)) = 𝑦)
12	8, 9, 10, 11	mp3an 1457	. . . 4 ⊢ (2nd ‘(1st ‘⟨𝑥, 𝑦, 𝑧⟩)) = 𝑦
13	7, 12	eqtr2di 2782	. . 3 ⊢ (𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ → 𝑦 = (2nd ‘(1st ‘𝐴)))
14		sbceq1a 3784	. . 3 ⊢ (𝑦 = (2nd ‘(1st ‘𝐴)) → ([(2nd ‘𝐴) / 𝑧]𝜑 ↔ [(2nd ‘(1st ‘𝐴)) / 𝑦][(2nd ‘𝐴) / 𝑧]𝜑))
15	13, 14	syl 17	. 2 ⊢ (𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ → ([(2nd ‘𝐴) / 𝑧]𝜑 ↔ [(2nd ‘(1st ‘𝐴)) / 𝑦][(2nd ‘𝐴) / 𝑧]𝜑))
16		2fveq3 6901	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ → (1st ‘(1st ‘𝐴)) = (1st ‘(1st ‘⟨𝑥, 𝑦, 𝑧⟩)))
17		ot1stg 8008	. . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (1st ‘(1st ‘⟨𝑥, 𝑦, 𝑧⟩)) = 𝑥)
18	8, 9, 10, 17	mp3an 1457	. . . 4 ⊢ (1st ‘(1st ‘⟨𝑥, 𝑦, 𝑧⟩)) = 𝑥
19	16, 18	eqtr2di 2782	. . 3 ⊢ (𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ → 𝑥 = (1st ‘(1st ‘𝐴)))
20		sbceq1a 3784	. . 3 ⊢ (𝑥 = (1st ‘(1st ‘𝐴)) → ([(2nd ‘(1st ‘𝐴)) / 𝑦][(2nd ‘𝐴) / 𝑧]𝜑 ↔ [(1st ‘(1st ‘𝐴)) / 𝑥][(2nd ‘(1st ‘𝐴)) / 𝑦][(2nd ‘𝐴) / 𝑧]𝜑))
21	19, 20	syl 17	. 2 ⊢ (𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ → ([(2nd ‘(1st ‘𝐴)) / 𝑦][(2nd ‘𝐴) / 𝑧]𝜑 ↔ [(1st ‘(1st ‘𝐴)) / 𝑥][(2nd ‘(1st ‘𝐴)) / 𝑦][(2nd ‘𝐴) / 𝑧]𝜑))
22	6, 15, 21	3bitrrd 305	1 ⊢ (𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ → ([(1st ‘(1st ‘𝐴)) / 𝑥][(2nd ‘(1st ‘𝐴)) / 𝑦][(2nd ‘𝐴) / 𝑧]𝜑 ↔ 𝜑))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1533   ∈ wcel 2098  Vcvv 3461  [wsbc 3773  ⟨cotp 4638  ‘cfv 6549  1^st c1st 7992  2^nd c2nd 7993

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429  ax-un 7741

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-sbc 3774  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-ot 4639  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-iota 6501  df-fun 6551  df-fv 6557  df-1st 7994  df-2nd 7995

This theorem is referenced by:  ralxp3es  8144  frpoins3xp3g  8146