Theorem bj-vtoclg1f 34797

Description: Reprove vtoclg1f 3473 from bj-vtoclg1f1 34796. This removes dependency on ax-ext 2706, df-cleq 2726 and df-v 3403. Use bj-vtoclg1fv 34798 instead when sufficient (in particular when 𝑉 is substituted for V). (Contributed by BJ, 14-Sep-2019.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
bj-vtoclg1f.nf	⊢ Ⅎ𝑥𝜓
bj-vtoclg1f.maj	⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓))
bj-vtoclg1f.min	⊢ 𝜑

Assertion

Ref	Expression
bj-vtoclg1f	⊢ (𝐴 ∈ 𝑉 → 𝜓)

Distinct variable group:   𝑥,𝐴

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

Proof of Theorem bj-vtoclg1f

Step	Hyp	Ref	Expression
1		bj-elisset 34755	. 2 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴)
2		bj-vtoclg1f.nf	. . 3 ⊢ Ⅎ𝑥𝜓
3		bj-vtoclg1f.maj	. . 3 ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓))
4		bj-vtoclg1f.min	. . 3 ⊢ 𝜑
5	2, 3, 4	bj-exlimmpi 34791	. 2 ⊢ (∃𝑥 𝑥 = 𝐴 → 𝜓)
6	1, 5	syl 17	1 ⊢ (𝐴 ∈ 𝑉 → 𝜓)

Syntax hints:   → wi 4   = wceq 1543  ∃wex 1787  Ⅎwnf 1791   ∈ wcel 2110

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-12 2175

This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1546  df-ex 1788  df-nf 1792  df-sb 2071  df-clab 2713  df-clel 2812

This theorem is referenced by: (None)