Theorem qimeqallhalf-P7 975

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

The reverse implication is also true if '𝑥' is ENF in either '𝜑' (see qimeqalla-P7 1050) or '𝜓' (see qimeqallb-P7r 1052).

Assertion

Ref	Expression
qimeqallhalf-P7	⊢ ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑 → 𝜓))

Proof of Theorem qimeqallhalf-P7

Step	Hyp	Ref	Expression
1		ndnfrex1-P7.8 833	. . 3 ⊢ Ⅎ𝑥∃𝑥𝜑
2		ndnfrall1-P7.7 832	. . 3 ⊢ Ⅎ𝑥∀𝑥𝜓
3	1, 2	ndnfrim-P7.3.RC 876	. 2 ⊢ Ⅎ𝑥(∃𝑥𝜑 → ∀𝑥𝜓)
4		exi-P7.CL 952	. . 3 ⊢ (𝜑 → ∃𝑥𝜑)
5		alle-P7.CL 942	. . 3 ⊢ (∀𝑥𝜓 → 𝜓)
6	4, 5	imsubd-P3.28c.RC 272	. 2 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜓) → (𝜑 → 𝜓))
7	3, 6	alli-P7 947	1 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑 → 𝜓))

Syntax hints:  ∀wff-forall 8   → wff-imp 10  ∃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  ax-L5 17  ax-L6 18  ax-L7 19  ax-L10 27  ax-L11 28  ax-L12 29

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  df-nfree-D6.1 682  df-psub-D6.2 716

This theorem is referenced by:  qimeqallb-P7  976  qimeqallhalf-P7r  1049  qimeqalla-P7  1050