Theorem alloverimex-P7 948

Description: Alternate version of alloverim-P7 970 that produces existential quantifiers. †

Hypothesis

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

Assertion

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

Proof of Theorem alloverimex-P7

Step	Hyp	Ref	Expression
1		alloverimex-P7.1	. 2 ⊢ (𝛾 → ∀𝑥(𝜑 → 𝜓))
2		axL4ex-P7 946	. 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  ∃wff-exists 595

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  949  alloverimex-P7r  1028  alloverimex-P7r.RC  1029  alloverimex-P7r.GENV  1031  dalloverimex-P7  1033