Theorem dalloverim-P5 590

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

Hypothesis

Ref	Expression
dalloverim-P5.1	⊢ (𝛾 → ∀𝑥(𝜑 → (𝜓 → 𝜒)))

Assertion

Ref	Expression
dalloverim-P5	⊢ (𝛾 → (∀𝑥𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒)))

Proof of Theorem dalloverim-P5

Step	Hyp	Ref	Expression
1		dalloverim-P5.1	. . . . 5 ⊢ (𝛾 → ∀𝑥(𝜑 → (𝜓 → 𝜒)))
2	1	alloverim-P5 588	. . . 4 ⊢ (𝛾 → (∀𝑥𝜑 → ∀𝑥(𝜓 → 𝜒)))
3	2	rcp-IMPIME1 527	. . 3 ⊢ ((𝛾 ∧ ∀𝑥𝜑) → ∀𝑥(𝜓 → 𝜒))
4	3	alloverim-P5 588	. 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-L4 16

This theorem depends on definitions:  df-bi-D2.1 107  df-and-D2.2 133  df-true-D2.4 155

This theorem is referenced by:  dalloverim-P5.RC  591