Theorem alloverim-P5 588

Description: '∀𝑥' Distributes Over '→'.

This is the deductive form of ax-L4 16.

Hypothesis

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

Assertion

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

Proof of Theorem alloverim-P5

Step	Hyp	Ref	Expression
1		alloverim-P5.1	. 2 ⊢ (𝛾 → ∀𝑥(𝜑 → 𝜓))
2		ax-L4 16	. . 3 ⊢ (∀𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))
3	2	rcp-NDIMP0addall 207	. 2 ⊢ (𝛾 → (∀𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓)))
4	1, 3	ndime-P3.6 171	1 ⊢ (𝛾 → (∀𝑥𝜑 → ∀𝑥𝜓))

Syntax hints:  ∀wff-forall 8   → wff-imp 10

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:  alloverim-P5.RC  589  dalloverim-P5  590  alloverimex-P5  601  dalloverimex-P5  605  alloverim-P5.GENV  621  alloverim-P5.GENF  747