Theorem qimeqallhalf-P5 609

Description: Partial Quantified Implication Equivalence Law ( ( E → U ) → U ).

The reverse implication is only true when '𝑥' either does not occur in '𝜑' (qimeqallav-P5 618) or '𝜓' (qimeqallbv-P5 620), or does not occur free in '𝜑' (qimeqalla-P6 699) or '𝜓' (qimeqallb-P6 701).

Assertion

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

Proof of Theorem qimeqallhalf-P5

Step	Hyp	Ref	Expression
1		impoe-P4.4a.CL 379	. . . . 5 ⊢ (¬ 𝜑 → (𝜑 → 𝜓))
2	1	alloverim-P5.RC.GEN 592	. . . 4 ⊢ (∀𝑥 ¬ 𝜑 → ∀𝑥(𝜑 → 𝜓))
3		allnegex-P5 597	. . . 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	alloverim-P5.RC.GEN 592	. . 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:  qimeqallav-P5  618  qimeqallbv-P5  620  nfrim-P6  689  qimeqalla-P6  699  qimeqallb-P6  701  nfrimd-P6  815