Theorem alloverimex-P7.GENF 949

Description: alloverimex-P7 948 with generalization augmentation (non-freeness condition). †

Hypotheses

Ref	Expression
alloverimex-P7.GENF.1	⊢ Ⅎ𝑥𝛾
alloverimex-P7.GENF.2	⊢ (𝛾 → (𝜑 → 𝜓))

Assertion

Ref	Expression
alloverimex-P7.GENF	⊢ (𝛾 → (∃𝑥𝜑 → ∃𝑥𝜓))

Proof of Theorem alloverimex-P7.GENF

Step	Hyp	Ref	Expression
1		alloverimex-P7.GENF.1	. . 3 ⊢ Ⅎ𝑥𝛾
2		alloverimex-P7.GENF.2	. . 3 ⊢ (𝛾 → (𝜑 → 𝜓))
3	1, 2	alli-P7 947	. 2 ⊢ (𝛾 → ∀𝑥(𝜑 → 𝜓))
4	3	alloverimex-P7 948	1 ⊢ (𝛾 → (∃𝑥𝜑 → ∃𝑥𝜓))

Syntax hints:   → wff-imp 10  ∃wff-exists 595  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-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:  alloverimex-P7.GENF.RC  950  exia-P7  953  alloverimex-P7r.GENF  1030  subex-P7  1042