Theorem spcgv 2912

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 2392	. 2 ⊢ Ⅎ𝑥𝐴
2		nfv 1581	. 2 ⊢ Ⅎ𝑥𝜓
3		spcgv.1	. 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
4	1, 2, 3	spcgf 2907	1 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥𝜑 → 𝜓))

Syntax hints:   → wi 4   ↔ wb 105  ∀wal 1400   = wceq 1402   ∈ wcel 2209

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823

This theorem is referenced by:  spcv  2919  mob2  3006  intss1  3980  dfiin2g  4040  exmidsssnc  4335  exmid1stab  4340  frirrg  4490  frind  4492  alxfr  4602  elirr  4683  en2lp  4696  tfisi  4729  mptfvex  5785  tfrcl  6625  rdgisucinc  6646  frecabex  6659  fisseneq  7232  mkvprop  7488  exmidfodomrlemr  7544  exmidfodomrlemrALT  7545  acfun  7553  exmidmotap  7617  ccfunen  7620  zfz1isolem1  11270  zfz1iso  11271  uniopn  15025  pw1nct  16947  sbthom  16976