Theorem r19.30 3063

Description: Restricted quantifier version of 19.30 1798. (Contributed by Scott Fenton, 25-Feb-2011.)

Assertion

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

Proof of Theorem r19.30

Step	Hyp	Ref	Expression
1		ralim 2932	. 2 ⊢ (∀𝑥 ∈ 𝐴 (¬ 𝜓 → 𝜑) → (∀𝑥 ∈ 𝐴 ¬ 𝜓 → ∀𝑥 ∈ 𝐴 𝜑))
2		orcom 401	. . . 4 ⊢ ((𝜑 ∨ 𝜓) ↔ (𝜓 ∨ 𝜑))
3		df-or 384	. . . 4 ⊢ ((𝜓 ∨ 𝜑) ↔ (¬ 𝜓 → 𝜑))
4	2, 3	bitri 263	. . 3 ⊢ ((𝜑 ∨ 𝜓) ↔ (¬ 𝜓 → 𝜑))
5	4	ralbii 2963	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ↔ ∀𝑥 ∈ 𝐴 (¬ 𝜓 → 𝜑))
6		orcom 401	. . 3 ⊢ ((∀𝑥 ∈ 𝐴 𝜑 ∨ ¬ ∀𝑥 ∈ 𝐴 ¬ 𝜓) ↔ (¬ ∀𝑥 ∈ 𝐴 ¬ 𝜓 ∨ ∀𝑥 ∈ 𝐴 𝜑))
7		dfrex2 2979	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝜓 ↔ ¬ ∀𝑥 ∈ 𝐴 ¬ 𝜓)
8	7	orbi2i 540	. . 3 ⊢ ((∀𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∨ ¬ ∀𝑥 ∈ 𝐴 ¬ 𝜓))
9		imor 427	. . 3 ⊢ ((∀𝑥 ∈ 𝐴 ¬ 𝜓 → ∀𝑥 ∈ 𝐴 𝜑) ↔ (¬ ∀𝑥 ∈ 𝐴 ¬ 𝜓 ∨ ∀𝑥 ∈ 𝐴 𝜑))
10	6, 8, 9	3bitr4i 291	. 2 ⊢ ((∀𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓) ↔ (∀𝑥 ∈ 𝐴 ¬ 𝜓 → ∀𝑥 ∈ 𝐴 𝜑))
11	1, 5, 10	3imtr4i 280	1 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) → (∀𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 382  ∀wral 2896  ∃wrex 2897

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-ex 1696  df-ral 2901  df-rex 2902

This theorem is referenced by:  disjunsn  28789  esumcvg  29475