Theorem spcgv 3552

Description: Rule of specialization, using implicit substitution. Compare Theorem 7.3 of [Quine] p. 44. (Contributed by NM, 22-Jun-1994.) Avoid ax-10 2147, ax-11 2163. (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 3463	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		id 22	. . 3 ⊢ (𝐴 ∈ V → 𝐴 ∈ V)
3		spcgv.1	. . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
4	3	adantl 481	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝑥 = 𝐴) → (𝜑 ↔ 𝜓))
5	2, 4	spcdv 3550	. 2 ⊢ (𝐴 ∈ V → (∀𝑥𝜑 → 𝜓))
6	1, 5	syl 17	1 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥𝜑 → 𝜓))

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1540   = wceq 1542   ∈ wcel 2114  Vcvv 3442

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3444

This theorem is referenced by:  spcv  3561  mob2  3675  sbceqal  3804  intss1  4920  dfiin2g  4988  alxfr  5356  funmo  6518  isofrlem  7298  tfisi  7813  limomss  7825  nnlim  7834  f1oweALT  7928  pssnn  9107  findcard3  9197  frmin  9675  ttukeylem1  10433  rami  16957  ramcl  16971  islbs3  21127  mplsubglem  21971  mpllsslem  21972  uniopn  22858  chlimi  31328  iinabrex  32662  dfon2lem3  36005  dfon2lem8  36010  neificl  38033  hashnexinj  42527  ismrcd1  43084  mnuop23d  44651  relpfrlem  45338  modelaxreplem2  45364