Theorem bj-vtoclf 37405

Description: Remove dependency on ax-ext 2735, df-clab 2742 and df-cleq 2755 (and df-sb 2092 and df-v 3457) from vtoclf 3531. (Contributed by BJ, 6-Oct-2019.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
bj-vtoclf.nf	⊢ Ⅎ𝑥𝜓
bj-vtoclf.s	⊢ 𝐴 ∈ 𝑉
bj-vtoclf.maj	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
bj-vtoclf.min	⊢ 𝜑

Assertion

Ref	Expression
bj-vtoclf	⊢ 𝜓

Distinct variable groups:   𝑥,𝐴   𝑥,𝑉

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

Proof of Theorem bj-vtoclf

Step	Hyp	Ref	Expression
1		bj-vtoclf.nf	. . 3 ⊢ Ⅎ𝑥𝜓
2		bj-vtoclf.s	. . . . 5 ⊢ 𝐴 ∈ 𝑉
3	2	bj-issetiv 37367	. . . 4 ⊢ ∃𝑥 𝑥 = 𝐴
4		bj-vtoclf.maj	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
5	4	biimpd 231	. . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓))
6	3, 5	eximii 1858	. . 3 ⊢ ∃𝑥(𝜑 → 𝜓)
7	1, 6	19.36i 2267	. 2 ⊢ (∀𝑥𝜑 → 𝜓)
8		bj-vtoclf.min	. 2 ⊢ 𝜑
9	7, 8	mpg 1818	1 ⊢ 𝜓

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1561  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:  bj-vtocl  37406