Theorem vtoclg1f 3582

Description: Version of vtoclgf 3581 with one nonfreeness hypothesis replaced with a disjoint variable condition, thus avoiding dependency on ax-10 2141 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 2826	. 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 1537  ∃wex 1777  Ⅎwnf 1781   ∈ wcel 2108

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-12 2178

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-clel 2819

This theorem is referenced by:  ceqsexg  3666  elabgOLD  3691  mob  3739  opeliunxp2  5863  fvopab5  7062  opeliunxp2f  8251  fprodsplit1f  16038  cnextfvval  24094  dvfsumlem2  26087  dvfsumlem2OLD  26088  dvfsumlem4  26090  bnj981  34926  dmrelrnrel  45133  fmul01  45501  fmuldfeq  45504  fmul01lt1lem1  45505  fprodcnlem  45520  stoweidlem3  45924  stoweidlem26  45947  stoweidlem31  45952  stoweidlem43  45964  stoweidlem51  45972  fourierdlem86  46113  fourierdlem89  46116  fourierdlem91  46118  salpreimagelt  46628  salpreimalegt  46630