Theorem sbcop 5434

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 5433	. . 3 ⊢ ([𝑎 / 𝑥]𝜓 ↔ [⟨𝑎, 𝑦⟩ / 𝑧]𝜑)
3	2	sbcbii 3794	. 2 ⊢ ([𝑏 / 𝑦][𝑎 / 𝑥]𝜓 ↔ [𝑏 / 𝑦][⟨𝑎, 𝑦⟩ / 𝑧]𝜑)
4		sbcnestgw 4372	. . 3 ⊢ (𝑏 ∈ V → ([𝑏 / 𝑦][⟨𝑎, 𝑦⟩ / 𝑧]𝜑 ↔ [⦋𝑏 / 𝑦⦌⟨𝑎, 𝑦⟩ / 𝑧]𝜑))
5	4	elv 3442	. 2 ⊢ ([𝑏 / 𝑦][⟨𝑎, 𝑦⟩ / 𝑧]𝜑 ↔ [⦋𝑏 / 𝑦⦌⟨𝑎, 𝑦⟩ / 𝑧]𝜑)
6		csbopg 4844	. . . . 5 ⊢ (𝑏 ∈ V → ⦋𝑏 / 𝑦⦌⟨𝑎, 𝑦⟩ = ⟨⦋𝑏 / 𝑦⦌𝑎, ⦋𝑏 / 𝑦⦌𝑦⟩)
7	6	elv 3442	. . . 4 ⊢ ⦋𝑏 / 𝑦⦌⟨𝑎, 𝑦⟩ = ⟨⦋𝑏 / 𝑦⦌𝑎, ⦋𝑏 / 𝑦⦌𝑦⟩
8		vex 3441	. . . . . 6 ⊢ 𝑏 ∈ V
9	8	csbconstgi 3867	. . . . 5 ⊢ ⦋𝑏 / 𝑦⦌𝑎 = 𝑎
10	8	csbvargi 4384	. . . . 5 ⊢ ⦋𝑏 / 𝑦⦌𝑦 = 𝑏
11	9, 10	opeq12i 4831	. . . 4 ⊢ ⟨⦋𝑏 / 𝑦⦌𝑎, ⦋𝑏 / 𝑦⦌𝑦⟩ = ⟨𝑎, 𝑏⟩
12	7, 11	eqtri 2756	. . 3 ⊢ ⦋𝑏 / 𝑦⦌⟨𝑎, 𝑦⟩ = ⟨𝑎, 𝑏⟩
13		dfsbcq 3739	. . 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 297	1 ⊢ ([𝑏 / 𝑦][𝑎 / 𝑥]𝜓 ↔ [⟨𝑎, 𝑏⟩ / 𝑧]𝜑)

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1541  Vcvv 3437  [wsbc 3737  ⦋csb 3846  ⟨cop 4583

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584

This theorem is referenced by:  reuop  6248