Theorem qimeqallav-P5-L1 617

Description: Lemma for qimeqallav-P5 618.

Assertion

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

Distinct variable group:   𝜑,𝑥

Proof of Theorem qimeqallav-P5-L1

Step	Hyp	Ref	Expression
1		axL5ex-P5 613	. . 3 ⊢ (∃𝑥𝜑 → 𝜑)
2	1	rcp-NDIMP0addall 207	. 2 ⊢ (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → 𝜑))
3		ax-L5 17	. . 3 ⊢ (𝜑 → ∀𝑥𝜑)
4	3	rcp-NDIMP0addall 207	. 2 ⊢ (∀𝑥(𝜑 → 𝜓) → (𝜑 → ∀𝑥𝜑))
5		ax-L4 16	. 2 ⊢ (∀𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))
6	2, 4, 5	dsyl-P3.25 261	1 ⊢ (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓))

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-L4 16  ax-L5 17

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