Theorem csbiedOLD 3882

Description: Obsolete version of csbied 3881 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 1916	. 2 ⊢ Ⅎ𝑥𝜑
2		nfcvd 2905	. 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 3874	1 ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐶)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1540   ∈ wcel 2105  ⦋csb 3843

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-v 3443  df-sbc 3728  df-csb 3844

This theorem is referenced by: (None)