Theorem bj-vtoclg1fv 37409

Description: Version of bj-vtoclg1f 37408 with a disjoint variable condition on 𝑥, 𝑉. This removes dependency on df-sb 2092 and df-clab 2742. Prefer its use over bj-vtoclg1f 37408 when sufficient (in particular when 𝑉 is substituted for V). (Contributed by BJ, 14-Sep-2019.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
bj-vtoclg1fv.nf	⊢ Ⅎ𝑥𝜓
bj-vtoclg1fv.maj	⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓))
bj-vtoclg1fv.min	⊢ 𝜑

Assertion

Ref	Expression
bj-vtoclg1fv	⊢ (𝐴 ∈ 𝑉 → 𝜓)

Distinct variable groups:   𝑥,𝐴   𝑥,𝑉

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem bj-vtoclg1fv

Step	Hyp	Ref	Expression
1		elissetv 2844	. 2 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴)
2		bj-vtoclg1fv.nf	. . 3 ⊢ Ⅎ𝑥𝜓
3		bj-vtoclg1fv.maj	. . 3 ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓))
4		bj-vtoclg1fv.min	. . 3 ⊢ 𝜑
5	2, 3, 4	bj-exlimmpi 37402	. 2 ⊢ (∃𝑥 𝑥 = 𝐴 → 𝜓)
6	1, 5	syl 17	1 ⊢ (𝐴 ∈ 𝑉 → 𝜓)

Syntax hints:   → wi 4   = wceq 1561  ∃wex 1800  Ⅎwnf 1804   ∈ wcel 2143

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-12 2213

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1801  df-nf 1805  df-clel 2838

This theorem is referenced by: (None)