Theorem bj-vtoclgf 11106

Description: Weakening two hypotheses of vtoclgf 2671. (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 11105	. 2 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (𝐴 ∈ 𝑉 → 𝜓))
5		bj-vtoclgf.maj	. 2 ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓))
6	4, 5	mpg 1383	1 ⊢ (𝐴 ∈ 𝑉 → 𝜓)

Syntax hints:   → wi 4   = wceq 1287  Ⅎwnf 1392   ∈ wcel 1436  Ⅎwnfc 2212

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 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067

This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2617

This theorem is referenced by:  elabgf2  11110