Theorem vtoclg1f 3527

Description: Version of vtoclgf 3526 with one nonfreeness hypothesis replaced with a disjoint variable condition, thus avoiding dependency on ax-10 2147 and ax-11 2163. (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		elisset 2819	. 2 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴)
2		vtoclg1f.nf	. . 3 ⊢ Ⅎ𝑥𝜓
3		vtoclg1f.min	. . . 4 ⊢ 𝜑
4		vtoclg1f.maj	. . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
5	3, 4	mpbii 233	. . 3 ⊢ (𝑥 = 𝐴 → 𝜓)
6	2, 5	exlimi 2225	. 2 ⊢ (∃𝑥 𝑥 = 𝐴 → 𝜓)
7	1, 6	syl 17	1 ⊢ (𝐴 ∈ 𝑉 → 𝜓)

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1542  ∃wex 1781  Ⅎwnf 1785   ∈ wcel 2114

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-12 2185

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2716  df-clel 2812

This theorem is referenced by:  ceqsexg  3608  mob  3676  opeliunxp2  5788  fvopab5  6976  opeliunxp2f  8154  fprodsplit1f  15917  cnextfvval  24013  dvfsumlem2  25993  dvfsumlem2OLD  25994  dvfsumlem4  25996  bnj981  35108  dmrelrnrel  45537  fmul01  45893  fmuldfeq  45896  fmul01lt1lem1  45897  fprodcnlem  45912  stoweidlem3  46314  stoweidlem26  46337  stoweidlem31  46342  stoweidlem43  46354  stoweidlem51  46362  fourierdlem86  46503  fourierdlem89  46506  fourierdlem91  46508  sge0f1o  46693  salpreimagelt  47018  salpreimalegt  47020