Theorem exia-P7 953

Description: Introduce Existential Quantifier as Antecedent. †

Hypotheses

Ref	Expression
exia-P7.1	⊢ Ⅎ𝑥𝛾
exia-P7.2	⊢ Ⅎ𝑥𝜓
exia-P7.3	⊢ (𝛾 → (𝜑 → 𝜓))

Assertion

Ref	Expression
exia-P7	⊢ (𝛾 → (∃𝑥𝜑 → 𝜓))

Proof of Theorem exia-P7

Step	Hyp	Ref	Expression
1		exia-P7.1	. . 3 ⊢ Ⅎ𝑥𝛾
2		exia-P7.3	. . 3 ⊢ (𝛾 → (𝜑 → 𝜓))
3	1, 2	alloverimex-P7.GENF 949	. 2 ⊢ (𝛾 → (∃𝑥𝜑 → ∃𝑥𝜓))
4		exia-P7.2	. . . 4 ⊢ Ⅎ𝑥𝜓
5	4	nfrexgen-P7.CL 932	. . 3 ⊢ (∃𝑥𝜓 → 𝜓)
6	5	rcp-NDIMP0addall 207	. 2 ⊢ (𝛾 → (∃𝑥𝜓 → 𝜓))
7	3, 6	syl-P3.24 259	1 ⊢ (𝛾 → (∃𝑥𝜑 → 𝜓))

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:  exia-P7.RC  954  exe-P7  955  dfnfreealtonlyif-P7  966  exia-P7r  1011  exia-P7r.VR1of2  1012  exia-P7r.VR2of2  1013  exia-P7r.VR12of2  1014