Theorem ra5 3141

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 1637. (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 2533	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝜑 → 𝜓)))
2		bi2.04 248	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 → (𝜑 → 𝜓)) ↔ (𝜑 → (𝑥 ∈ 𝐴 → 𝜓)))
3	2	albii 1523	. . . 4 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → (𝜑 → 𝜓)) ↔ ∀𝑥(𝜑 → (𝑥 ∈ 𝐴 → 𝜓)))
4	1, 3	bitri 184	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ ∀𝑥(𝜑 → (𝑥 ∈ 𝐴 → 𝜓)))
5		ra5.1	. . . 4 ⊢ Ⅎ𝑥𝜑
6	5	stdpc5 1637	. . 3 ⊢ (∀𝑥(𝜑 → (𝑥 ∈ 𝐴 → 𝜓)) → (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 → 𝜓)))
7	4, 6	sylbi 121	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 → 𝜓)))
8		df-ral 2533	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜓))
9	7, 8	imbitrrdi 162	1 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (𝜑 → ∀𝑥 ∈ 𝐴 𝜓))

This proof depends on syntax axioms:   → wi 4  ∀wal 1400  Ⅎwnf 1513   ∈ wcel 2209  ∀wral 2528

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-gen 1502  ax-4 1563  ax-ial 1587  ax-i5r 1588

This proof depends on definitions:  df-bi 117  df-nf 1514  df-ral 2533

This theorem is used by: (None)