Theorem sbcoteq1a 8030

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 6858	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ → (2nd ‘𝐴) = (2nd ‘⟨𝑥, 𝑦, 𝑧⟩))
2		ot3rdg 7984	. . . . 5 ⊢ (𝑧 ∈ V → (2nd ‘⟨𝑥, 𝑦, 𝑧⟩) = 𝑧)
3	2	elv 3452	. . . 4 ⊢ (2nd ‘⟨𝑥, 𝑦, 𝑧⟩) = 𝑧
4	1, 3	eqtr2di 2781	. . 3 ⊢ (𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ → 𝑧 = (2nd ‘𝐴))
5		sbceq1a 3764	. . 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 6863	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ → (2nd ‘(1st ‘𝐴)) = (2nd ‘(1st ‘⟨𝑥, 𝑦, 𝑧⟩)))
8		vex 3451	. . . . 5 ⊢ 𝑥 ∈ V
9		vex 3451	. . . . 5 ⊢ 𝑦 ∈ V
10		vex 3451	. . . . 5 ⊢ 𝑧 ∈ V
11		ot2ndg 7983	. . . . 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 2781	. . 3 ⊢ (𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ → 𝑦 = (2nd ‘(1st ‘𝐴)))
14		sbceq1a 3764	. . 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 6863	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ → (1st ‘(1st ‘𝐴)) = (1st ‘(1st ‘⟨𝑥, 𝑦, 𝑧⟩)))
17		ot1stg 7982	. . . . 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 2781	. . 3 ⊢ (𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ → 𝑥 = (1st ‘(1st ‘𝐴)))
20		sbceq1a 3764	. . 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 1540   ∈ wcel 2109  Vcvv 3447  [wsbc 3753  ⟨cotp 4597  ‘cfv 6511  1^st c1st 7966  2^nd c2nd 7967

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-ot 4598  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-iota 6464  df-fun 6513  df-fv 6519  df-1st 7968  df-2nd 7969

This theorem is referenced by:  ralxp3es  8118  frpoins3xp3g  8120