Theorem vtoclg1f 2789

Description: Version of vtoclgf 2788 with one nonfreeness hypothesis replaced with a disjoint variable condition, thus avoiding dependency on ax-11 1499 and ax-13 2143. (Contributed by BJ, 1-May-2019.)

Hypotheses

Ref	Expression
vtoclg1f.nf	⊢ Ⅎ𝑥𝜓
vtoclg1f.maj	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
vtoclg1f.min	⊢ 𝜑

Assertion

Ref	Expression
vtoclg1f	⊢ (𝐴 ∈ 𝑉 → 𝜓)

Distinct variable group:   𝑥,𝐴

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

Proof of Theorem vtoclg1f

Step	Hyp	Ref	Expression
1		elex 2741	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		isset 2736	. . 3 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
3		vtoclg1f.nf	. . . 4 ⊢ Ⅎ𝑥𝜓
4		vtoclg1f.min	. . . . 5 ⊢ 𝜑
5		vtoclg1f.maj	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
6	4, 5	mpbii 147	. . . 4 ⊢ (𝑥 = 𝐴 → 𝜓)
7	3, 6	exlimi 1587	. . 3 ⊢ (∃𝑥 𝑥 = 𝐴 → 𝜓)
8	2, 7	sylbi 120	. 2 ⊢ (𝐴 ∈ V → 𝜓)
9	1, 8	syl 14	1 ⊢ (𝐴 ∈ 𝑉 → 𝜓)

Syntax hints:   → wi 4   ↔ wb 104   = wceq 1348  Ⅎwnf 1453  ∃wex 1485   ∈ wcel 2141  Vcvv 2730

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-v 2732

This theorem is referenced by:  opeliunxp2f  6217  summodclem2a  11344  fprodsplit1f  11597