Theorem alloverim-P5.GENF 747

Description: alloverim-P5 588 with Generalization (non-freeness condition).

Hypotheses

Ref	Expression
alloverim-P5.GENF.1	⊢ Ⅎ𝑥𝛾
alloverim-P5.GENF.2	⊢ (𝛾 → (𝜑 → 𝜓))

Assertion

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

Proof of Theorem alloverim-P5.GENF

Step	Hyp	Ref	Expression
1		alloverim-P5.GENF.1	. . 3 ⊢ Ⅎ𝑥𝛾
2		alloverim-P5.GENF.2	. . 3 ⊢ (𝛾 → (𝜑 → 𝜓))
3	1, 2	allic-P6 745	. 2 ⊢ (𝛾 → ∀𝑥(𝜑 → 𝜓))
4	3	alloverim-P5 588	1 ⊢ (𝛾 → (∀𝑥𝜑 → ∀𝑥𝜓))

Syntax hints:  ∀wff-forall 8   → wff-imp 10  Ⅎwff-nfree 681

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-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

This theorem is referenced by:  suball-P6  753  nfrall2d-P6  819