Theorem spcv 2846

Description: Rule of specialization, using implicit substitution. (Contributed by NM, 22-Jun-1994.)

Hypotheses

Ref	Expression
spcv.1	⊢ 𝐴 ∈ V
spcv.2	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
spcv	⊢ (∀𝑥𝜑 → 𝜓)

Distinct variable groups:   𝑥,𝐴   𝜓,𝑥

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem spcv

Step	Hyp	Ref	Expression
1		spcv.1	. 2 ⊢ 𝐴 ∈ V
2		spcv.2	. . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
3	2	spcgv 2839	. 2 ⊢ (𝐴 ∈ V → (∀𝑥𝜑 → 𝜓))
4	1, 3	ax-mp 5	1 ⊢ (∀𝑥𝜑 → 𝜓)

Syntax hints:   → wi 4   ↔ wb 105  ∀wal 1362   = wceq 1364   ∈ wcel 2160  Vcvv 2752

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754

This theorem is referenced by:  morex  2936  exmidexmid  4211  exmidsssn  4217  exmidel  4220  rext  4230  ontr2exmid  4539  regexmidlem1  4547  reg2exmid  4550  relop  4792  disjxp1  6256  rdgtfr  6394  ssfiexmid  6899  domfiexmid  6901  diffitest  6910  findcard  6911  exmidpw2en  6935  fiintim  6952  fisseneq  6955  finomni  7163  exmidomni  7165  exmidlpo  7166  exmidunben  12472  bj-d0clsepcl  15115  bj-inf2vnlem1  15160  subctctexmid  15189