Theorem spcgv 2906

Description: Rule of specialization, using implicit substitution. Compare Theorem 7.3 of [Quine] p. 44. (Contributed by NM, 22-Jun-1994.)

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		nfcv 2386	. 2 ⊢ Ⅎ𝑥𝐴
2		nfv 1577	. 2 ⊢ Ⅎ𝑥𝜓
3		spcgv.1	. 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
4	1, 2, 3	spcgf 2901	1 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥𝜑 → 𝜓))

Syntax hints:   → wi 4   ↔ wb 105  ∀wal 1396   = wceq 1398   ∈ wcel 2205

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817

This theorem is referenced by:  spcv  2913  mob2  3000  intss1  3969  dfiin2g  4029  exmidsssnc  4321  exmid1stab  4326  frirrg  4476  frind  4478  alxfr  4587  elirr  4668  en2lp  4681  tfisi  4714  mptfvex  5768  tfrcl  6608  rdgisucinc  6629  frecabex  6642  fisseneq  7208  mkvprop  7462  exmidfodomrlemr  7518  exmidfodomrlemrALT  7519  acfun  7527  exmidmotap  7591  ccfunen  7594  zfz1isolem1  11237  zfz1iso  11238  uniopn  14992  pw1nct  16903  sbthom  16932