Theorem bj-vtoclf 37645

Description: Remove dependency on ax-ext 2734, df-clab 2741 and df-cleq 2754 (and df-sb 2100 and df-v 3455) from vtoclf 3528. (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 37607	. . . 4 ⊢ ∃𝑥 𝑥 = 𝐴
4		bj-vtoclf.maj	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
5	4	biimpd 232	. . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓))
6	3, 5	eximii 1870	. . 3 ⊢ ∃𝑥(𝜑 → 𝜓)
7	1, 6	19.36i 2269	. 2 ⊢ (∀𝑥𝜑 → 𝜓)
8		bj-vtoclf.min	. 2 ⊢ 𝜑
9	7, 8	mpg 1830	1 ⊢ 𝜓

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570  Ⅎwnf 1816   ∈ wcel 2145

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-12 2215

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-nf 1817  df-clel 2837

This theorem is used by:  bj-vtocl  37646