Theorem spcgv 3550

Description: Rule of specialization, using implicit substitution. Compare Theorem 7.3 of [Quine] p. 44. (Contributed by NM, 22-Jun-1994.) Avoid ax-10 2177, ax-11 2193. (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 3471	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		elex 3471	. . 3 ⊢ (𝐴 ∈ V → 𝐴 ∈ V)
3		spcgv.1	. . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
4	3	adantl 487	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝑥 = 𝐴) → (𝜑 ↔ 𝜓))
5	2, 4	spcdv 3548	. 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 1567   = wceq 1569   ∈ wcel 2144  Vcvv 3450

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452

This theorem is used by:  spcv  3559  mob2  3672  sbceqal  3799  intss1  4922  alxfr  5368  funmo  6554  isofrlem  7348  tfisi  7870  limomss  7882  nnlim  7891  f1oweALT  7984  pssnn  9184  findcard3  9274  frmin  9753  ttukeylem1  10587  rami  17193  ramcl  17207  islbs3  21433  mplsubglem  22306  mpllsslem  22307  uniopn  23215  chlimi  31836  iinabrex  33163  dfon2lem3  36547  dfon2lem8  36552  neificl  38688  hashnexinj  43179  ismrcd1  43709  mnuop23d  45250  relpfrlem  45943  modelaxreplem2  45969