Theorem sbcoteq1a 8000

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 6834	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ → (2nd ‘𝐴) = (2nd ‘⟨𝑥, 𝑦, 𝑧⟩))
2		ot3rdg 7954	. . . . 5 ⊢ (𝑧 ∈ V → (2nd ‘⟨𝑥, 𝑦, 𝑧⟩) = 𝑧)
3	2	elv 3437	. . . 4 ⊢ (2nd ‘⟨𝑥, 𝑦, 𝑧⟩) = 𝑧
4	1, 3	eqtr2di 2792	. . 3 ⊢ (𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ → 𝑧 = (2nd ‘𝐴))
5		sbceq1a 3741	. . 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 6839	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ → (2nd ‘(1st ‘𝐴)) = (2nd ‘(1st ‘⟨𝑥, 𝑦, 𝑧⟩)))
8		vex 3436	. . . . 5 ⊢ 𝑥 ∈ V
9		vex 3436	. . . . 5 ⊢ 𝑦 ∈ V
10		vex 3436	. . . . 5 ⊢ 𝑧 ∈ V
11		ot2ndg 7953	. . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (2nd ‘(1st ‘⟨𝑥, 𝑦, 𝑧⟩)) = 𝑦)
12	8, 9, 10, 11	mp3an 1469	. . . 4 ⊢ (2nd ‘(1st ‘⟨𝑥, 𝑦, 𝑧⟩)) = 𝑦
13	7, 12	eqtr2di 2792	. . 3 ⊢ (𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ → 𝑦 = (2nd ‘(1st ‘𝐴)))
14		sbceq1a 3741	. . 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 6839	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ → (1st ‘(1st ‘𝐴)) = (1st ‘(1st ‘⟨𝑥, 𝑦, 𝑧⟩)))
17		ot1stg 7952	. . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (1st ‘(1st ‘⟨𝑥, 𝑦, 𝑧⟩)) = 𝑥)
18	8, 9, 10, 17	mp3an 1469	. . . 4 ⊢ (1st ‘(1st ‘⟨𝑥, 𝑦, 𝑧⟩)) = 𝑥
19	16, 18	eqtr2di 2792	. . 3 ⊢ (𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ → 𝑥 = (1st ‘(1st ‘𝐴)))
20		sbceq1a 3741	. . 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 307	1 ⊢ (𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ → ([(1st ‘(1st ‘𝐴)) / 𝑥][(2nd ‘(1st ‘𝐴)) / 𝑦][(2nd ‘𝐴) / 𝑧]𝜑 ↔ 𝜑))

Syntax hints:   → wi 4   ↔ wb 207   = wceq 1547   ∈ wcel 2119  Vcvv 3432  [wsbc 3730  ⟨cotp 4570  ‘cfv 6492  1^st c1st 7936  2^nd c2nd 7937

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-sbc 3731  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-ot 4571  df-uni 4846  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-iota 6448  df-fun 6494  df-fv 6500  df-1st 7938  df-2nd 7939

This theorem is referenced by:  ralxp3es  8086  frpoins3xp3g  8088