Theorem sbcop 5465

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 5464	. . 3 ⊢ ([𝑎 / 𝑥]𝜓 ↔ [⟨𝑎, 𝑦⟩ / 𝑧]𝜑)
3	2	sbcbii 3795	. 2 ⊢ ([𝑏 / 𝑦][𝑎 / 𝑥]𝜓 ↔ [𝑏 / 𝑦][⟨𝑎, 𝑦⟩ / 𝑧]𝜑)
4		sbcnestgw 4381	. . 3 ⊢ (𝑏 ∈ V → ([𝑏 / 𝑦][⟨𝑎, 𝑦⟩ / 𝑧]𝜑 ↔ [⦋𝑏 / 𝑦⦌⟨𝑎, 𝑦⟩ / 𝑧]𝜑))
5	4	elv 3455	. 2 ⊢ ([𝑏 / 𝑦][⟨𝑎, 𝑦⟩ / 𝑧]𝜑 ↔ [⦋𝑏 / 𝑦⦌⟨𝑎, 𝑦⟩ / 𝑧]𝜑)
6		csbopg 4851	. . . . 5 ⊢ (𝑏 ∈ V → ⦋𝑏 / 𝑦⦌⟨𝑎, 𝑦⟩ = ⟨⦋𝑏 / 𝑦⦌𝑎, ⦋𝑏 / 𝑦⦌𝑦⟩)
7	6	elv 3455	. . . 4 ⊢ ⦋𝑏 / 𝑦⦌⟨𝑎, 𝑦⟩ = ⟨⦋𝑏 / 𝑦⦌𝑎, ⦋𝑏 / 𝑦⦌𝑦⟩
8		vex 3454	. . . . . 6 ⊢ 𝑏 ∈ V
9	8	csbconstgi 3868	. . . . 5 ⊢ ⦋𝑏 / 𝑦⦌𝑎 = 𝑎
10	8	csbvargi 4393	. . . . 5 ⊢ ⦋𝑏 / 𝑦⦌𝑦 = 𝑏
11	9, 10	opeq12i 4838	. . . 4 ⊢ ⟨⦋𝑏 / 𝑦⦌𝑎, ⦋𝑏 / 𝑦⦌𝑦⟩ = ⟨𝑎, 𝑏⟩
12	7, 11	eqtri 2783	. . 3 ⊢ ⦋𝑏 / 𝑦⦌⟨𝑎, 𝑦⟩ = ⟨𝑎, 𝑏⟩
13		dfsbcq 3741	. . 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 300	1 ⊢ ([𝑏 / 𝑦][𝑎 / 𝑥]𝜓 ↔ [⟨𝑎, 𝑏⟩ / 𝑧]𝜑)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570  Vcvv 3450  [wsbc 3739  ⦋csb 3847  ⟨cop 4590

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-pr 5398

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-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  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

This theorem is used by:  reuop  6291