Theorem qimeqex-P5-L1 610

Description: Lemma for qimeqex-P5 612.

Assertion

Ref	Expression
qimeqex-P5-L1	⊢ ((∀𝑥𝜑 → ∃𝑥𝜓) → ∃𝑥(𝜑 → 𝜓))

Proof of Theorem qimeqex-P5-L1

Step	Hyp	Ref	Expression
1		impoe-P4.4a.CL 379	. . . . 5 ⊢ (¬ 𝜑 → (𝜑 → 𝜓))
2	1	alloverimex-P5.RC.GEN 603	. . . 4 ⊢ (∃𝑥 ¬ 𝜑 → ∃𝑥(𝜑 → 𝜓))
3		exnegall-P5 598	. . . 4 ⊢ (∃𝑥 ¬ 𝜑 ↔ ¬ ∀𝑥𝜑)
4	2, 3	subiml2-P4.RC 541	. . 3 ⊢ (¬ ∀𝑥𝜑 → ∃𝑥(𝜑 → 𝜓))
5		axL1-P3.21.CL 253	. . . 4 ⊢ (𝜓 → (𝜑 → 𝜓))
6	5	alloverimex-P5.RC.GEN 603	. . 3 ⊢ (∃𝑥𝜓 → ∃𝑥(𝜑 → 𝜓))
7	4, 6	joinimandinc3-P4.RC 579	. 2 ⊢ ((¬ ∀𝑥𝜑 ∨ ∃𝑥𝜓) → ∃𝑥(𝜑 → 𝜓))
8		imasor-P4.32a 487	. . 3 ⊢ ((∀𝑥𝜑 → ∃𝑥𝜓) ↔ (¬ ∀𝑥𝜑 ∨ ∃𝑥𝜓))
9	8	bisym-P3.33b.RC 299	. 2 ⊢ ((¬ ∀𝑥𝜑 ∨ ∃𝑥𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓))
10	7, 9	subiml2-P4.RC 541	1 ⊢ ((∀𝑥𝜑 → ∃𝑥𝜓) → ∃𝑥(𝜑 → 𝜓))

Syntax hints:  ∀wff-forall 8  ¬ wff-neg 9   → wff-imp 10   ∨ wff-or 144  ∃wff-exists 595

This theorem was proved from axioms:  ax-L1 11  ax-L2 12  ax-L3 13  ax-MP 14  ax-GEN 15  ax-L4 16

This theorem depends on definitions:  df-bi-D2.1 107  df-and-D2.2 133  df-or-D2.3 145  df-true-D2.4 155  df-rcp-AND3 161  df-exists-D5.1 596

This theorem is referenced by:  qimeqex-P5  612