Theorem ralim 2489

Description: Distribution of restricted quantification over implication. (Contributed by NM, 9-Feb-1997.)

Assertion

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

Proof of Theorem ralim

Step	Hyp	Ref	Expression
1		df-ral 2419	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝜑 → 𝜓)))
2		ax-2 7	. . . 4 ⊢ ((𝑥 ∈ 𝐴 → (𝜑 → 𝜓)) → ((𝑥 ∈ 𝐴 → 𝜑) → (𝑥 ∈ 𝐴 → 𝜓)))
3	2	al2imi 1434	. . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → (𝜑 → 𝜓)) → (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) → ∀𝑥(𝑥 ∈ 𝐴 → 𝜓)))
4	1, 3	sylbi 120	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) → ∀𝑥(𝑥 ∈ 𝐴 → 𝜓)))
5		df-ral 2419	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
6		df-ral 2419	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜓))
7	4, 5, 6	3imtr4g 204	1 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 𝜓))

Syntax hints:   → wi 4  ∀wal 1329   ∈ wcel 1480  ∀wral 2414

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425

This theorem depends on definitions:  df-bi 116  df-ral 2419

This theorem is referenced by:  ral2imi  2495  trint  4036  peano2  4504  mpteqb  5504  mptelixpg  6621  lbzbi  9401  r19.29uz  10757  alzdvds  11541