Theorem bj-vtoclg 37874

Description: A version of vtoclg 3517 with an additional disjoint variable condition (which is removable if we allow use of df-clab 2739, see bj-vtoclg1f 37872), which requires fewer axioms (i.e., removes dependency on ax-6 1999, ax-7 2040, ax-9 2154, ax-12 2212, ax-ext 2732, df-clab 2739, df-cleq 2752, df-v 3452). (Contributed by BJ, 2-Jul-2022.) (Proof modification is discouraged.)

Hypotheses

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

Assertion

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

Distinct variable groups:   𝑥,𝐴   𝑥,𝑉   𝜓,𝑥

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem bj-vtoclg

Step	Hyp	Ref	Expression
1		elissetv 2841	. 2 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴)
2		bj-vtoclg.maj	. . 3 ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓))
3		bj-vtoclg.min	. . 3 ⊢ 𝜑
4	2, 3	bj-exlimvmpi 37865	. 2 ⊢ (∃𝑥 𝑥 = 𝐴 → 𝜓)
5	1, 4	syl 18	1 ⊢ (𝐴 ∈ 𝑉 → 𝜓)

This proof depends on syntax axioms:   → wi 4   = wceq 1569  ∃wex 1811   ∈ wcel 2144

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-clel 2835

This theorem is used by:  bj-sepg  37878