Theorem vtoclg1f 3530

Description: Version of vtoclgf 3529 with one nonfreeness hypothesis replaced with a disjoint variable condition, thus avoiding dependency on ax-10 2178 and ax-11 2194. (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 2842	. 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 236	. . 3 ⊢ (𝑥 = 𝐴 → 𝜓)
6	2, 5	exlimi 2253	. 2 ⊢ (∃𝑥 𝑥 = 𝐴 → 𝜓)
7	1, 6	syl 18	1 ⊢ (𝐴 ∈ 𝑉 → 𝜓)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570  ∃wex 1812  Ⅎwnf 1816   ∈ wcel 2145

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-12 2213

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2739  df-clel 2835

This theorem is used by:  ceqsexg  3607  mob  3675  opeliunxp2  5812  fvopab5  7016  opeliunxp2f  8206  fprodsplit1f  16110  cnextfvval  24331  dvfsumlem2  26294  dvfsumlem4  26296  bnj981  35500  dmrelrnrel  46154  fmul01  46508  fmuldfeq  46511  fmul01lt1lem1  46512  fprodcnlem  46527  stoweidlem3  46929  stoweidlem26  46952  stoweidlem31  46957  stoweidlem43  46969  stoweidlem51  46977  fourierdlem86  47118  fourierdlem89  47121  fourierdlem91  47123  sge0f1o  47308  salpreimagelt  47633  salpreimalegt  47635