Theorem dfnfreealtif-P7 964

Description: Necessary Condition for (i.e. "If" part of) Alternate ENF Definition. †

Assertion

Ref	Expression
dfnfreealtif-P7	⊢ ((∃𝑥𝜑 → ∀𝑥𝜑) → Ⅎ𝑥𝜑)

Proof of Theorem dfnfreealtif-P7

Step	Hyp	Ref	Expression
1		ndnfrex1-P7.8 833	. . . 4 ⊢ Ⅎ𝑥∃𝑥𝜑
2		ndnfrall1-P7.7 832	. . . 4 ⊢ Ⅎ𝑥∀𝑥𝜑
3	1, 2	ndnfrim-P7.3.RC 876	. . 3 ⊢ Ⅎ𝑥(∃𝑥𝜑 → ∀𝑥𝜑)
4		exi-P7.CL 952	. . . 4 ⊢ (𝜑 → ∃𝑥𝜑)
5	4	imsubl-P3.28a.RC 268	. . 3 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜑) → (𝜑 → ∀𝑥𝜑))
6	3, 5	alli-P7 947	. 2 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜑) → ∀𝑥(𝜑 → ∀𝑥𝜑))
7		gennfrcl-P7 963	. 2 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → Ⅎ𝑥𝜑)
8	6, 7	syl-P3.24.RC 260	1 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜑) → Ⅎ𝑥𝜑)

Syntax hints:  ∀wff-forall 8   → wff-imp 10  ∃wff-exists 595  Ⅎwff-nfree 681

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:  dfnfreealt-P7  967  dfnfreeint-P7  969  nfrnegconv-P8  1110