Theorem dalloverim-P7 1022

Description: Double '∀𝑥' Distribution Over '→'. †

Hypothesis

Ref	Expression
dalloverim-P7.1	⊢ (𝛾 → ∀𝑥(𝜑 → (𝜓 → 𝜒)))

Assertion

Ref	Expression
dalloverim-P7	⊢ (𝛾 → (∀𝑥𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒)))

Proof of Theorem dalloverim-P7

Step	Hyp	Ref	Expression
1		dalloverim-P7.1	. . . . 5 ⊢ (𝛾 → ∀𝑥(𝜑 → (𝜓 → 𝜒)))
2	1	alloverim-P7 970	. . . 4 ⊢ (𝛾 → (∀𝑥𝜑 → ∀𝑥(𝜓 → 𝜒)))
3	2	import-P3.34a.RC 306	. . 3 ⊢ ((𝛾 ∧ ∀𝑥𝜑) → ∀𝑥(𝜓 → 𝜒))
4	3	alloverim-P7 970	. 2 ⊢ ((𝛾 ∧ ∀𝑥𝜑) → (∀𝑥𝜓 → ∀𝑥𝜒))
5	4	rcp-NDIMI2 224	1 ⊢ (𝛾 → (∀𝑥𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒)))

Syntax hints:  ∀wff-forall 8   → wff-imp 10   ∧ wff-and 132

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:  dalloverim-P7.RC  1023  dalloverim-P7.CL  1024  dalloverim-P7.GENF  1025  dalloverim-P7.GENV  1026