Theorem ra5 2846

Description: Restricted quantifier version of Axiom 5 of [Mendelson] p. 69. This is an axiom of a predicate calculus for a restricted domain. Compare the unrestricted stdpc5 1476. (Contributed by NM, 16-Jan-2004.)

Hypothesis

Ref	Expression
ra5.1	⊢ Ⅎ𝑥𝜑

Assertion

Ref	Expression
ra5	⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (𝜑 → ∀𝑥 ∈ 𝐴 𝜓))

Proof of Theorem ra5

Step	Hyp	Ref	Expression
1		df-ral 2311	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝜑 → 𝜓)))
2		bi2.04 237	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 → (𝜑 → 𝜓)) ↔ (𝜑 → (𝑥 ∈ 𝐴 → 𝜓)))
3	2	albii 1359	. . . 4 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → (𝜑 → 𝜓)) ↔ ∀𝑥(𝜑 → (𝑥 ∈ 𝐴 → 𝜓)))
4	1, 3	bitri 173	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ ∀𝑥(𝜑 → (𝑥 ∈ 𝐴 → 𝜓)))
5		ra5.1	. . . 4 ⊢ Ⅎ𝑥𝜑
6	5	stdpc5 1476	. . 3 ⊢ (∀𝑥(𝜑 → (𝑥 ∈ 𝐴 → 𝜓)) → (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 → 𝜓)))
7	4, 6	sylbi 114	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 → 𝜓)))
8		df-ral 2311	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜓))
9	7, 8	syl6ibr 151	1 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (𝜑 → ∀𝑥 ∈ 𝐴 𝜓))

Syntax hints:   → wi 4  ∀wal 1241  Ⅎwnf 1349   ∈ wcel 1393  ∀wral 2306

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-4 1400  ax-ial 1427  ax-i5r 1428

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-ral 2311

This theorem is referenced by: (None)