Theorem bj-vtoclg 34239

Description: A version of vtoclg 3567 with an additional disjoint variable condition (which is removable if we allow use of df-clab 2800, see bj-vtoclg1f 34237), which requires fewer axioms (i.e., removes dependency on ax-6 1970, ax-7 2015, ax-9 2124, ax-12 2177, ax-ext 2793, df-clab 2800, df-cleq 2814, df-v 3496). (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		bj-elissetv 34194	. 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 34230	. 2 ⊢ (∃𝑥 𝑥 = 𝐴 → 𝜓)
5	1, 4	syl 17	1 ⊢ (𝐴 ∈ 𝑉 → 𝜓)

Syntax hints:   → wi 4   = wceq 1537  ∃wex 1780   ∈ wcel 2114

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 1911  ax-6 1970  ax-7 2015  ax-8 2116

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781  df-clel 2893

This theorem is referenced by:  bj-zfauscl  34246