Theorem exia-P6 746

Description: Introduction of Existential Quantifier as Antecedent (non-freeness condition).

This proposition is equivalent to the '∃' elimination rule in the natural deduction system.

Hypotheses

Ref	Expression
exia-P6.1	⊢ Ⅎ𝑥𝜓
exia-P6.2	⊢ (𝜑 → 𝜓)

Assertion

Ref	Expression
exia-P6	⊢ (∃𝑥𝜑 → 𝜓)

Proof of Theorem exia-P6

Step	Hyp	Ref	Expression
1		exia-P6.2	. . 3 ⊢ (𝜑 → 𝜓)
2	1	alloverimex-P5.RC.GEN 603	. 2 ⊢ (∃𝑥𝜑 → ∃𝑥𝜓)
3		exia-P6.1	. . 3 ⊢ Ⅎ𝑥𝜓
4	3	nfrexgen-P6 735	. 2 ⊢ (∃𝑥𝜓 → 𝜓)
5	2, 4	syl-P3.24.RC 260	1 ⊢ (∃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:  spliteq-P6  776  splitelof-P6  778