Theorem sbcop 5429

Description: The proper substitution of an ordered pair for a setvar variable corresponds to a proper substitution of each of its components. (Contributed by AV, 8-Apr-2023.)

Hypothesis

Ref	Expression
sbcop.z	⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
sbcop	⊢ ([𝑏 / 𝑦][𝑎 / 𝑥]𝜓 ↔ [⟨𝑎, 𝑏⟩ / 𝑧]𝜑)

Distinct variable groups:   𝑥,𝑎,𝑦,𝑧   𝜑,𝑥,𝑦   𝜓,𝑧   𝑥,𝑏,𝑦,𝑧

Allowed substitution hints:   𝜑(𝑧,𝑎,𝑏)   𝜓(𝑥,𝑦,𝑎,𝑏)

Proof of Theorem sbcop

Step	Hyp	Ref	Expression
1		sbcop.z	. . . 4 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ 𝜓))
2	1	sbcop1 5428	. . 3 ⊢ ([𝑎 / 𝑥]𝜓 ↔ [⟨𝑎, 𝑦⟩ / 𝑧]𝜑)
3	2	sbcbii 3779	. 2 ⊢ ([𝑏 / 𝑦][𝑎 / 𝑥]𝜓 ↔ [𝑏 / 𝑦][⟨𝑎, 𝑦⟩ / 𝑧]𝜑)
4		sbcnestgw 4351	. . 3 ⊢ (𝑏 ∈ V → ([𝑏 / 𝑦][⟨𝑎, 𝑦⟩ / 𝑧]𝜑 ↔ [⦋𝑏 / 𝑦⦌⟨𝑎, 𝑦⟩ / 𝑧]𝜑))
5	4	elv 3436	. 2 ⊢ ([𝑏 / 𝑦][⟨𝑎, 𝑦⟩ / 𝑧]𝜑 ↔ [⦋𝑏 / 𝑦⦌⟨𝑎, 𝑦⟩ / 𝑧]𝜑)
6		csbopg 4822	. . . . 5 ⊢ (𝑏 ∈ V → ⦋𝑏 / 𝑦⦌⟨𝑎, 𝑦⟩ = ⟨⦋𝑏 / 𝑦⦌𝑎, ⦋𝑏 / 𝑦⦌𝑦⟩)
7	6	elv 3436	. . . 4 ⊢ ⦋𝑏 / 𝑦⦌⟨𝑎, 𝑦⟩ = ⟨⦋𝑏 / 𝑦⦌𝑎, ⦋𝑏 / 𝑦⦌𝑦⟩
8		vex 3435	. . . . . 6 ⊢ 𝑏 ∈ V
9	8	csbconstgi 3852	. . . . 5 ⊢ ⦋𝑏 / 𝑦⦌𝑎 = 𝑎
10	8	csbvargi 4363	. . . . 5 ⊢ ⦋𝑏 / 𝑦⦌𝑦 = 𝑏
11	9, 10	opeq12i 4809	. . . 4 ⊢ ⟨⦋𝑏 / 𝑦⦌𝑎, ⦋𝑏 / 𝑦⦌𝑦⟩ = ⟨𝑎, 𝑏⟩
12	7, 11	eqtri 2762	. . 3 ⊢ ⦋𝑏 / 𝑦⦌⟨𝑎, 𝑦⟩ = ⟨𝑎, 𝑏⟩
13		dfsbcq 3725	. . 3 ⊢ (⦋𝑏 / 𝑦⦌⟨𝑎, 𝑦⟩ = ⟨𝑎, 𝑏⟩ → ([⦋𝑏 / 𝑦⦌⟨𝑎, 𝑦⟩ / 𝑧]𝜑 ↔ [⟨𝑎, 𝑏⟩ / 𝑧]𝜑))
14	12, 13	ax-mp 5	. 2 ⊢ ([⦋𝑏 / 𝑦⦌⟨𝑎, 𝑦⟩ / 𝑧]𝜑 ↔ [⟨𝑎, 𝑏⟩ / 𝑧]𝜑)
15	3, 5, 14	3bitri 298	1 ⊢ ([𝑏 / 𝑦][𝑎 / 𝑥]𝜓 ↔ [⟨𝑎, 𝑏⟩ / 𝑧]𝜑)

Syntax hints:   → wi 4   ↔ wb 207   = wceq 1547  Vcvv 3431  [wsbc 3723  ⦋csb 3831  ⟨cop 4561

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 2711  ax-sep 5218  ax-pr 5362

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-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562

This theorem is referenced by:  reuop  6244