Theorem bj-vtoclg1f1 37407

Description: The FOL content of vtoclg1f 3536 (hence not using ax-ext 2735, df-cleq 2755, df-nfc 2912, df-v 3457). Note the weakened "major" hypothesis and the disjoint variable condition between 𝑥 and 𝐴 (needed since the nonfreeness quantifier for classes is not available without ax-ext 2735; as a byproduct, this dispenses with ax-11 2192 and ax-13 2404). (Contributed by BJ, 30-Apr-2019.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
bj-vtoclg1f1.nf	⊢ Ⅎ𝑥𝜓
bj-vtoclg1f1.maj	⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓))
bj-vtoclg1f1.min	⊢ 𝜑

Assertion

Ref	Expression
bj-vtoclg1f1	⊢ (∃𝑦 𝑦 = 𝐴 → 𝜓)

Distinct variable groups:   𝑥,𝐴   𝑦,𝐴

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

Proof of Theorem bj-vtoclg1f1

Step	Hyp	Ref	Expression
1		iseqsetv-clel 2842	. 2 ⊢ (∃𝑦 𝑦 = 𝐴 ↔ ∃𝑥 𝑥 = 𝐴)
2		bj-vtoclg1f1.nf	. . 3 ⊢ Ⅎ𝑥𝜓
3		bj-vtoclg1f1.maj	. . 3 ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓))
4		bj-vtoclg1f1.min	. . 3 ⊢ 𝜑
5	2, 3, 4	bj-exlimmpi 37402	. 2 ⊢ (∃𝑥 𝑥 = 𝐴 → 𝜓)
6	1, 5	sylbi 219	1 ⊢ (∃𝑦 𝑦 = 𝐴 → 𝜓)

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

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-tru 1564  df-ex 1801  df-nf 1805  df-sb 2092  df-clab 2742  df-clel 2838

This theorem is referenced by: (None)