Theorem cla4gv 1396

Description: Rule of specialization with implicit substitution. Compare Theorem 7.3 of [Quine] p. 44.

Hypothesis

Ref	Expression
cla4gv.1	⊢ (x = A → (φ ↔ ψ))

Assertion

Ref	Expression
cla4gv	⊢ (A ∈ B → (∀xφ → ψ))

Distinct variable group(s):   ψ,x   x,A

Proof of Theorem cla4gv

Step	Hyp	Ref	Expression
1		ax-17 925	. 2 ⊢ (y ∈ A → ∀x y ∈ A)
2		ax-17 925	. 2 ⊢ (ψ → ∀xψ)
3		cla4gv.1	. 2 ⊢ (x = A → (φ ↔ ψ))
4	1, 2, 3	cla4gf 1394	1 ⊢ (A ∈ B → (∀xφ → ψ))

Syntax hints:   → wi 2   ↔ wb 127  ∀wal 672   = wceq 1091   ∈ wcel 1092

This theorem is referenced by:  cla4v 1400  rcla4v 1402  elinti 1974  intss1 1979  limomss 2378  nnlim 2385  isofrlem 2939  f1oweOLD 2944  pssnn 3428  chlim 5139

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-12 802  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349

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