Theorem vtoclg1f 3536

Description: Version of vtoclgf 3535 with one nonfreeness hypothesis replaced with a disjoint variable condition, thus avoiding dependency on ax-10 2142 and ax-11 2158. (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 2810	. 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 2218	. 2 ⊢ (∃𝑥 𝑥 = 𝐴 → 𝜓)
7	1, 6	syl 17	1 ⊢ (𝐴 ∈ 𝑉 → 𝜓)

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1540  ∃wex 1779  Ⅎwnf 1783   ∈ wcel 2109

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-12 2178

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-clel 2803

This theorem is referenced by:  ceqsexg  3619  mob  3688  opeliunxp2  5802  fvopab5  7001  opeliunxp2f  8189  fprodsplit1f  15956  cnextfvval  23952  dvfsumlem2  25933  dvfsumlem2OLD  25934  dvfsumlem4  25936  bnj981  34940  dmrelrnrel  45220  fmul01  45578  fmuldfeq  45581  fmul01lt1lem1  45582  fprodcnlem  45597  stoweidlem3  46001  stoweidlem26  46024  stoweidlem31  46029  stoweidlem43  46041  stoweidlem51  46049  fourierdlem86  46190  fourierdlem89  46193  fourierdlem91  46195  sge0f1o  46380  salpreimagelt  46705  salpreimalegt  46707