Theorem spcgv 2893

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 2374	. 2 ⊢ Ⅎ𝑥𝐴
2		nfv 1576	. 2 ⊢ Ⅎ𝑥𝜓
3		spcgv.1	. 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
4	1, 2, 3	spcgf 2888	1 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥𝜑 → 𝜓))

Syntax hints:   → wi 4   ↔ wb 105  ∀wal 1395   = wceq 1397   ∈ wcel 2202

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804

This theorem is referenced by:  spcv  2900  mob2  2986  intss1  3943  dfiin2g  4003  exmidsssnc  4293  exmid1stab  4298  frirrg  4447  frind  4449  alxfr  4558  elirr  4639  en2lp  4652  tfisi  4685  mptfvex  5732  tfrcl  6530  rdgisucinc  6551  frecabex  6564  fisseneq  7127  mkvprop  7357  exmidfodomrlemr  7413  exmidfodomrlemrALT  7414  acfun  7422  exmidmotap  7480  ccfunen  7483  zfz1isolem1  11105  zfz1iso  11106  uniopn  14728  pw1nct  16625  sbthom  16651