Theorem axL4-P7 945

Description: ax-L4 16 Derived from Natural Deduction Rules. †

Assertion

Ref	Expression
axL4-P7	⊢ (∀𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))

Proof of Theorem axL4-P7

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ndalle-P7.18.CL 909	. . . . . 6 ⊢ (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑)
2	1	rcp-NDIMP0addall 207	. . . . 5 ⊢ (∀𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑))
3		ndnfrall1-P7.7 832	. . . . . 6 ⊢ Ⅎ𝑥∀𝑥(𝜑 → 𝜓)
4		alle-P7.CL 942	. . . . . 6 ⊢ (∀𝑥(𝜑 → 𝜓) → (𝜑 → 𝜓))
5	3, 4	lemma-L7.02a 944	. . . . 5 ⊢ (∀𝑥(𝜑 → 𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
6	2, 5	syl-P3.24 259	. . . 4 ⊢ (∀𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → [𝑦 / 𝑥]𝜓))
7	6	import-P3.34a.RC 306	. . 3 ⊢ ((∀𝑥(𝜑 → 𝜓) ∧ ∀𝑥𝜑) → [𝑦 / 𝑥]𝜓)
8	7	ndalli-P7.17.VR12of2 866	. 2 ⊢ ((∀𝑥(𝜑 → 𝜓) ∧ ∀𝑥𝜑) → ∀𝑥𝜓)
9	8	rcp-NDIMI2 224	1 ⊢ (∀𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))

Syntax hints:  term-obj 1  ∀wff-forall 8   → wff-imp 10   ∧ wff-and 132  [wff-psub 714

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:  alli-P7  947  alloverim-P7  970  qimeqalla-P7  1050