Theorem sbcoteq1a 8048

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 6878	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ → (2nd ‘𝐴) = (2nd ‘⟨𝑥, 𝑦, 𝑧⟩))
2		ot3rdg 8002	. . . . 5 ⊢ (𝑧 ∈ V → (2nd ‘⟨𝑥, 𝑦, 𝑧⟩) = 𝑧)
3	2	elv 3455	. . . 4 ⊢ (2nd ‘⟨𝑥, 𝑦, 𝑧⟩) = 𝑧
4	1, 3	eqtr2di 2812	. . 3 ⊢ (𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ → 𝑧 = (2nd ‘𝐴))
5		sbceq1a 3750	. . 3 ⊢ (𝑧 = (2nd ‘𝐴) → (𝜑 ↔ [(2nd ‘𝐴) / 𝑧]𝜑))
6	4, 5	syl 18	. 2 ⊢ (𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ → (𝜑 ↔ [(2nd ‘𝐴) / 𝑧]𝜑))
7		2fveq3 6883	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ → (2nd ‘(1st ‘𝐴)) = (2nd ‘(1st ‘⟨𝑥, 𝑦, 𝑧⟩)))
8		vex 3454	. . . . 5 ⊢ 𝑥 ∈ V
9		vex 3454	. . . . 5 ⊢ 𝑦 ∈ V
10		vex 3454	. . . . 5 ⊢ 𝑧 ∈ V
11		ot2ndg 8001	. . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (2nd ‘(1st ‘⟨𝑥, 𝑦, 𝑧⟩)) = 𝑦)
12	8, 9, 10, 11	mp3an 1490	. . . 4 ⊢ (2nd ‘(1st ‘⟨𝑥, 𝑦, 𝑧⟩)) = 𝑦
13	7, 12	eqtr2di 2812	. . 3 ⊢ (𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ → 𝑦 = (2nd ‘(1st ‘𝐴)))
14		sbceq1a 3750	. . 3 ⊢ (𝑦 = (2nd ‘(1st ‘𝐴)) → ([(2nd ‘𝐴) / 𝑧]𝜑 ↔ [(2nd ‘(1st ‘𝐴)) / 𝑦][(2nd ‘𝐴) / 𝑧]𝜑))
15	13, 14	syl 18	. 2 ⊢ (𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ → ([(2nd ‘𝐴) / 𝑧]𝜑 ↔ [(2nd ‘(1st ‘𝐴)) / 𝑦][(2nd ‘𝐴) / 𝑧]𝜑))
16		2fveq3 6883	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ → (1st ‘(1st ‘𝐴)) = (1st ‘(1st ‘⟨𝑥, 𝑦, 𝑧⟩)))
17		ot1stg 8000	. . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (1st ‘(1st ‘⟨𝑥, 𝑦, 𝑧⟩)) = 𝑥)
18	8, 9, 10, 17	mp3an 1490	. . . 4 ⊢ (1st ‘(1st ‘⟨𝑥, 𝑦, 𝑧⟩)) = 𝑥
19	16, 18	eqtr2di 2812	. . 3 ⊢ (𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ → 𝑥 = (1st ‘(1st ‘𝐴)))
20		sbceq1a 3750	. . 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 18	. 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 309	1 ⊢ (𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ → ([(1st ‘(1st ‘𝐴)) / 𝑥][(2nd ‘(1st ‘𝐴)) / 𝑦][(2nd ‘𝐴) / 𝑧]𝜑 ↔ 𝜑))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2145  Vcvv 3450  [wsbc 3739  ⟨cotp 4592  ‘cfv 6533  1^st c1st 7984  2^nd c2nd 7985

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pr 5398  ax-un 7736

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3740  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-ot 4593  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-iota 6489  df-fun 6535  df-fv 6541  df-1st 7986  df-2nd 7987

This theorem is used by:  ralxp3es  8137  frpoins3xp3g  8139