Theorem exiad-P6 824

Description: Introduction of Existential Quantifier as Antecedent (deductive form).

Hypotheses

Ref	Expression
exiad-P6.1	⊢ Ⅎ𝑥𝛾
exiad-P6.2	⊢ (𝛾 → Ⅎ𝑥𝜓)
exiad-P6.3	⊢ (𝛾 → (𝜑 → 𝜓))

Assertion

Ref	Expression
exiad-P6	⊢ (𝛾 → (∃𝑥𝜑 → 𝜓))

Proof of Theorem exiad-P6

Step	Hyp	Ref	Expression
1		exiad-P6.1	. . 3 ⊢ Ⅎ𝑥𝛾
2		exiad-P6.3	. . 3 ⊢ (𝛾 → (𝜑 → 𝜓))
3	1, 2	alloverimex-P5.GENF 748	. 2 ⊢ (𝛾 → (∃𝑥𝜑 → ∃𝑥𝜓))
4		exiad-P6.2	. . . 4 ⊢ (𝛾 → Ⅎ𝑥𝜓)
5	4	qremexd-P6 823	. . 3 ⊢ (𝛾 → (∃𝑥𝜓 ↔ 𝜓))
6	5	ndbief-P3.14 179	. 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-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:  ndexe-P6  825