Theorem alloverim-P7 970

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

Hypothesis

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

Assertion

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

Proof of Theorem alloverim-P7

Step	Hyp	Ref	Expression
1		alloverim-P7.1	. 2 ⊢ (𝛾 → ∀𝑥(𝜑 → 𝜓))
2		axL4-P7 945	. 2 ⊢ (∀𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))
3	1, 2	syl-P3.24.RC 260	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-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:  alloverim-P7.GENF  971  alloverim-P7r  1017  alloverim-P7r.RC  1018  alloverim-P7r.GENV  1020  dalloverim-P7  1022  dalloverimex-P7  1033