Theorem vtoclg1f 3296

Description: Version of vtoclgf 3295 with one non-freeness hypothesis replaced with a dv condition, thus avoiding dependency on ax-11 2074 and ax-13 2282. (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 3243	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		isset 3238	. . 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 223	. . . 4 ⊢ (𝑥 = 𝐴 → 𝜓)
7	3, 6	exlimi 2124	. . 3 ⊢ (∃𝑥 𝑥 = 𝐴 → 𝜓)
8	2, 7	sylbi 207	. 2 ⊢ (𝐴 ∈ V → 𝜓)
9	1, 8	syl 17	1 ⊢ (𝐴 ∈ 𝑉 → 𝜓)

Syntax hints:   → wi 4   ↔ wb 196   = wceq 1523  ∃wex 1744  Ⅎwnf 1748   ∈ wcel 2030  Vcvv 3231

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-12 2087  ax-ext 2631

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-v 3233

This theorem is referenced by:  vtoclg  3297  ceqsexg  3365  mob  3421  opeliunxp2  5293  fvopab5  6349  opeliunxp2f  7381  cnextfvval  21916  dvfsumlem2  23835  dvfsumlem4  23837  bnj981  31146  dmrelrnrel  39733  fmul01  40130  fmuldfeq  40133  fmul01lt1lem1  40134  cncfiooicclem1  40424  stoweidlem3  40538  stoweidlem26  40561  stoweidlem31  40566  stoweidlem43  40578  stoweidlem51  40586  fourierdlem86  40727  fourierdlem89  40730  fourierdlem91  40732