Theorem spcgv 2824

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 2319	. 2 ⊢ Ⅎ𝑥𝐴
2		nfv 1528	. 2 ⊢ Ⅎ𝑥𝜓
3		spcgv.1	. 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
4	1, 2, 3	spcgf 2819	1 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥𝜑 → 𝜓))

Syntax hints:   → wi 4   ↔ wb 105  ∀wal 1351   = wceq 1353   ∈ wcel 2148

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739

This theorem is referenced by:  spcv  2831  mob2  2917  intss1  3858  dfiin2g  3918  exmidsssnc  4201  exmid1stab  4206  frirrg  4348  frind  4350  alxfr  4459  elirr  4538  en2lp  4551  tfisi  4584  mptfvex  5598  tfrcl  6360  rdgisucinc  6381  frecabex  6394  fisseneq  6926  mkvprop  7151  exmidfodomrlemr  7196  exmidfodomrlemrALT  7197  acfun  7201  exmidmotap  7255  ccfunen  7258  zfz1isolem1  10811  zfz1iso  10812  uniopn  13281  pw1nct  14523  sbthom  14545