Theorem spcgv 3555

Description: Rule of specialization, using implicit substitution. Compare Theorem 7.3 of [Quine] p. 44. (Contributed by NM, 22-Jun-1994.) Avoid ax-10 2176, ax-11 2192. (Revised by Wolf Lammen, 25-Aug-2023.)

Hypothesis

Ref	Expression
spcgv.1	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
spcgv	⊢ (𝐴 ∈ 𝑉 → (∀𝑥𝜑 → 𝜓))

Distinct variable groups:   𝜓,𝑥   𝑥,𝐴

Allowed substitution hints:   𝜑(𝑥)   𝑉(𝑥)

Proof of Theorem spcgv

Step	Hyp	Ref	Expression
1		elex 3476	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		elex 3476	. . 3 ⊢ (𝐴 ∈ V → 𝐴 ∈ V)
3		spcgv.1	. . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
4	3	adantl 486	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝑥 = 𝐴) → (𝜑 ↔ 𝜓))
5	2, 4	spcdv 3553	. 2 ⊢ (𝐴 ∈ V → (∀𝑥𝜑 → 𝜓))
6	1, 5	syl 18	1 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥𝜑 → 𝜓))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1568   = wceq 1570   ∈ wcel 2143  Vcvv 3455

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-v 3457

This theorem is used by:  spcv  3564  mob2  3678  sbceqal  3805  intss1  4928  alxfr  5378  funmo  6552  isofrlem  7338  tfisi  7851  limomss  7863  nnlim  7872  f1oweALT  7965  pssnn  9149  findcard3  9239  frmin  9717  ttukeylem1  10497  rami  17079  ramcl  17093  islbs3  21288  mplsubglem  22157  mpllsslem  22158  uniopn  23063  chlimi  31595  iinabrex  32923  dfon2lem3  36283  dfon2lem8  36288  neificl  38432  hashnexinj  42923  ismrcd1  43457  mnuop23d  45004  relpfrlem  45690  modelaxreplem2  45716