Theorem csbiedOLD 3837

Description: Obsolete version of csbied 3836 as of 15-Oct-2024. (Contributed by Mario Carneiro, 2-Dec-2014.) (Revised by Mario Carneiro, 13-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
csbiedOLD.1	⊢ (𝜑 → 𝐴 ∈ 𝑉)
csbiedOLD.2	⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 = 𝐶)

Assertion

Ref	Expression
csbiedOLD	⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐶)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝜑,𝑥

Allowed substitution hints:   𝐵(𝑥)   𝑉(𝑥)

Proof of Theorem csbiedOLD

Step	Hyp	Ref	Expression
1		nfv 1922	. 2 ⊢ Ⅎ𝑥𝜑
2		nfcvd 2898	. 2 ⊢ (𝜑 → Ⅎ𝑥𝐶)
3		csbiedOLD.1	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
4		csbiedOLD.2	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 = 𝐶)
5	1, 2, 3, 4	csbiedf 3829	1 ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐶)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1543   ∈ wcel 2112  ⦋csb 3798

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 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  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 2073  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-v 3400  df-sbc 3684  df-csb 3799

This theorem is referenced by: (None)