Theorem rax4 1471

Description: Restricted quantifier version of Axiom 4 of [Mendelson] p. 59. This is an axiom of a predicate calculus for a restricted domain. Compare the unrestricted stdpc4 869.

Assertion

Ref	Expression
rax4	⊢ (B ∈ A → (∀x ∈ A φ → [B / x]φ))

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

Proof of Theorem rax4

Step	Hyp	Ref	Expression
1		a4sbc 1444	. . . . 5 ⊢ (B ∈ A → (∀x(x ∈ A → φ) → [B / x](x ∈ A → φ)))
2		df-ral 1205	. . . . 5 ⊢ (∀x ∈ A φ ↔ ∀x(x ∈ A → φ))
3	1, 2	syl5ib 181	. . . 4 ⊢ (B ∈ A → (∀x ∈ A φ → [B / x](x ∈ A → φ)))
4		sbcim 1460	. . . 4 ⊢ (B ∈ A → ([B / x](x ∈ A → φ) ↔ ([B / x]x ∈ A → [B / x]φ)))
5	3, 4	sylibd 177	. . 3 ⊢ (B ∈ A → (∀x ∈ A φ → ([B / x]x ∈ A → [B / x]φ)))
6		sbcel1 1466	. . . 4 ⊢ (B ∈ A → ([B / x]x ∈ A ↔ B ∈ A))
7	6	imbi1d 465	. . 3 ⊢ (B ∈ A → (([B / x]x ∈ A → [B / x]φ) ↔ (B ∈ A → [B / x]φ)))
8	5, 7	sylibd 177	. 2 ⊢ (B ∈ A → (∀x ∈ A φ → (B ∈ A → [B / x]φ)))
9	8	pm2.43a 60	1 ⊢ (B ∈ A → (∀x ∈ A φ → [B / x]φ))

Syntax hints:   → wi 2  ∀wal 672   ∈ wcel 1092  ∀wral 1201  [wsbc 1440

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-10 800  ax-11 801  ax-12 802  ax-16 922  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-ral 1205  df-v 1349  df-sbc 1441

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