Theorem bj-vtoclgf 10586

Description: Weakening two hypotheses of vtoclgf 2657. (Contributed by BJ, 21-Nov-2019.)

Hypotheses

Ref	Expression
bj-vtoclgf.nf1	⊢ Ⅎ𝑥𝐴
bj-vtoclgf.nf2	⊢ Ⅎ𝑥𝜓
bj-vtoclgf.min	⊢ (𝑥 = 𝐴 → 𝜑)
bj-vtoclgf.maj	⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓))

Assertion

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

Proof of Theorem bj-vtoclgf

Step	Hyp	Ref	Expression
1		bj-vtoclgf.nf1	. . 3 ⊢ Ⅎ𝑥𝐴
2		bj-vtoclgf.nf2	. . 3 ⊢ Ⅎ𝑥𝜓
3		bj-vtoclgf.min	. . 3 ⊢ (𝑥 = 𝐴 → 𝜑)
4	1, 2, 3	bj-vtoclgft 10585	. 2 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (𝐴 ∈ 𝑉 → 𝜓))
5		bj-vtoclgf.maj	. 2 ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓))
6	4, 5	mpg 1380	1 ⊢ (𝐴 ∈ 𝑉 → 𝜓)

Syntax hints:   → wi 4   = wceq 1284  Ⅎwnf 1389   ∈ wcel 1433  Ⅎwnfc 2206

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603

This theorem is referenced by:  elabgf2  10590