Theorem sbc2ieOLD 3779

Description: Obsolete version of sbc2ie 3778 as of 12-Oct-2024. (Contributed by NM, 16-Dec-2008.) (Revised by Mario Carneiro, 19-Dec-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
sbc2ieOLD.1	⊢ 𝐴 ∈ V
sbc2ieOLD.2	⊢ 𝐵 ∈ V
sbc2ieOLD.3	⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
sbc2ieOLD	⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ 𝜓)

Distinct variable groups:   𝑥,𝑦,𝐴   𝑦,𝐵   𝜓,𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐵(𝑥)

Proof of Theorem sbc2ieOLD

Step	Hyp	Ref	Expression
1		sbc2ieOLD.1	. 2 ⊢ 𝐴 ∈ V
2		sbc2ieOLD.2	. 2 ⊢ 𝐵 ∈ V
3		nfv 1922	. . 3 ⊢ Ⅎ𝑥𝜓
4		nfv 1922	. . 3 ⊢ Ⅎ𝑦𝜓
5	2	nfth 1809	. . 3 ⊢ Ⅎ𝑥 𝐵 ∈ V
6		sbc2ieOLD.3	. . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓))
7	3, 4, 5, 6	sbc2iegf 3777	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ 𝜓))
8	1, 2, 7	mp2an 692	1 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ 𝜓)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1543   ∈ wcel 2110  Vcvv 3408  [wsbc 3694

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-12 2175  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-ex 1788  df-nf 1792  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-sbc 3695

This theorem is referenced by: (None)