Theorem r19.30OLD 3300

Description: Obsolete version of r19.30 3299 as of 18-Jun-2023. (Contributed by Scott Fenton, 25-Feb-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
r19.30OLD	⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) → (∀𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓))

Proof of Theorem r19.30OLD

Step	Hyp	Ref	Expression
1		ralim 3129	. 2 ⊢ (∀𝑥 ∈ 𝐴 (¬ 𝜓 → 𝜑) → (∀𝑥 ∈ 𝐴 ¬ 𝜓 → ∀𝑥 ∈ 𝐴 𝜑))
2		orcom 865	. . . 4 ⊢ ((𝜑 ∨ 𝜓) ↔ (𝜓 ∨ 𝜑))
3		df-or 843	. . . 4 ⊢ ((𝜓 ∨ 𝜑) ↔ (¬ 𝜓 → 𝜑))
4	2, 3	bitri 276	. . 3 ⊢ ((𝜑 ∨ 𝜓) ↔ (¬ 𝜓 → 𝜑))
5	4	ralbii 3132	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ↔ ∀𝑥 ∈ 𝐴 (¬ 𝜓 → 𝜑))
6		orcom 865	. . 3 ⊢ ((∀𝑥 ∈ 𝐴 𝜑 ∨ ¬ ∀𝑥 ∈ 𝐴 ¬ 𝜓) ↔ (¬ ∀𝑥 ∈ 𝐴 ¬ 𝜓 ∨ ∀𝑥 ∈ 𝐴 𝜑))
7		dfrex2 3203	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝜓 ↔ ¬ ∀𝑥 ∈ 𝐴 ¬ 𝜓)
8	7	orbi2i 907	. . 3 ⊢ ((∀𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∨ ¬ ∀𝑥 ∈ 𝐴 ¬ 𝜓))
9		imor 848	. . 3 ⊢ ((∀𝑥 ∈ 𝐴 ¬ 𝜓 → ∀𝑥 ∈ 𝐴 𝜑) ↔ (¬ ∀𝑥 ∈ 𝐴 ¬ 𝜓 ∨ ∀𝑥 ∈ 𝐴 𝜑))
10	6, 8, 9	3bitr4i 304	. 2 ⊢ ((∀𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓) ↔ (∀𝑥 ∈ 𝐴 ¬ 𝜓 → ∀𝑥 ∈ 𝐴 𝜑))
11	1, 5, 10	3imtr4i 293	1 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) → (∀𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 842  ∀wral 3105  ∃wrex 3106

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-ex 1762  df-ral 3110  df-rex 3111

This theorem is referenced by: (None)