Theorem sbcoteq1a 7983

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 6822	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ → (2nd ‘𝐴) = (2nd ‘⟨𝑥, 𝑦, 𝑧⟩))
2		ot3rdg 7937	. . . . 5 ⊢ (𝑧 ∈ V → (2nd ‘⟨𝑥, 𝑦, 𝑧⟩) = 𝑧)
3	2	elv 3441	. . . 4 ⊢ (2nd ‘⟨𝑥, 𝑦, 𝑧⟩) = 𝑧
4	1, 3	eqtr2di 2783	. . 3 ⊢ (𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ → 𝑧 = (2nd ‘𝐴))
5		sbceq1a 3752	. . 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 6827	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ → (2nd ‘(1st ‘𝐴)) = (2nd ‘(1st ‘⟨𝑥, 𝑦, 𝑧⟩)))
8		vex 3440	. . . . 5 ⊢ 𝑥 ∈ V
9		vex 3440	. . . . 5 ⊢ 𝑦 ∈ V
10		vex 3440	. . . . 5 ⊢ 𝑧 ∈ V
11		ot2ndg 7936	. . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (2nd ‘(1st ‘⟨𝑥, 𝑦, 𝑧⟩)) = 𝑦)
12	8, 9, 10, 11	mp3an 1463	. . . 4 ⊢ (2nd ‘(1st ‘⟨𝑥, 𝑦, 𝑧⟩)) = 𝑦
13	7, 12	eqtr2di 2783	. . 3 ⊢ (𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ → 𝑦 = (2nd ‘(1st ‘𝐴)))
14		sbceq1a 3752	. . 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 6827	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ → (1st ‘(1st ‘𝐴)) = (1st ‘(1st ‘⟨𝑥, 𝑦, 𝑧⟩)))
17		ot1stg 7935	. . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (1st ‘(1st ‘⟨𝑥, 𝑦, 𝑧⟩)) = 𝑥)
18	8, 9, 10, 17	mp3an 1463	. . . 4 ⊢ (1st ‘(1st ‘⟨𝑥, 𝑦, 𝑧⟩)) = 𝑥
19	16, 18	eqtr2di 2783	. . 3 ⊢ (𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ → 𝑥 = (1st ‘(1st ‘𝐴)))
20		sbceq1a 3752	. . 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 306	1 ⊢ (𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ → ([(1st ‘(1st ‘𝐴)) / 𝑥][(2nd ‘(1st ‘𝐴)) / 𝑦][(2nd ‘𝐴) / 𝑧]𝜑 ↔ 𝜑))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1541   ∈ wcel 2111  Vcvv 3436  [wsbc 3741  ⟨cotp 4584  ‘cfv 6481  1^st c1st 7919  2^nd c2nd 7920

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3742  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-ot 4585  df-uni 4860  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-iota 6437  df-fun 6483  df-fv 6489  df-1st 7921  df-2nd 7922

This theorem is referenced by:  ralxp3es  8069  frpoins3xp3g  8071