Theorem r19.32v 3190

Description: Restricted quantifier version of 19.32v 1938. (Contributed by NM, 25-Nov-2003.)

Assertion

Ref	Expression
r19.32v	⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ↔ (𝜑 ∨ ∀𝑥 ∈ 𝐴 𝜓))

Distinct variable group:   𝜑,𝑥

Allowed substitution hints:   𝜓(𝑥)   𝐴(𝑥)

Proof of Theorem r19.32v

Step	Hyp	Ref	Expression
1		r19.21v 3178	. 2 ⊢ (∀𝑥 ∈ 𝐴 (¬ 𝜑 → 𝜓) ↔ (¬ 𝜑 → ∀𝑥 ∈ 𝐴 𝜓))
2		df-or 848	. . 3 ⊢ ((𝜑 ∨ 𝜓) ↔ (¬ 𝜑 → 𝜓))
3	2	ralbii 3091	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ↔ ∀𝑥 ∈ 𝐴 (¬ 𝜑 → 𝜓))
4		df-or 848	. 2 ⊢ ((𝜑 ∨ ∀𝑥 ∈ 𝐴 𝜓) ↔ (¬ 𝜑 → ∀𝑥 ∈ 𝐴 𝜓))
5	1, 3, 4	3bitr4i 303	1 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ↔ (𝜑 ∨ ∀𝑥 ∈ 𝐴 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∨ wo 847  ∀wral 3059

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ral 3060

This theorem is referenced by:  2ralor  3229  iinun2  5078  iinuni  5103  axcontlem2  28995  axcontlem7  29000  disjnf  32590  lindslinindsimp2  48309