Theorem sbc2ieOLD 3872

Description: Obsolete version of sbc2ie 3871 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 1910	. . 3 ⊢ Ⅎ𝑥𝜓
4		nfv 1910	. . 3 ⊢ Ⅎ𝑦𝜓
5	2	nfth 1796	. . 3 ⊢ Ⅎ𝑥 𝐵 ∈ V
6		sbc2ieOLD.3	. . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓))
7	3, 4, 5, 6	sbc2iegf 3870	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ 𝜓))
8	1, 2, 7	mp2an 690	1 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ 𝜓)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1534   ∈ wcel 2100  Vcvv 3472  [wsbc 3788

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2102  ax-9 2110  ax-12 2170  ax-ext 2700

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-ex 1775  df-nf 1779  df-sb 2062  df-clab 2707  df-cleq 2721  df-clel 2806  df-sbc 3789

This theorem is referenced by: (None)