Theorem exiav-P5 615

Description: Introduction of Existential Quantifier as Antecedent (variable restriction).

'𝑥' may occur in '𝜑', but not '𝜓'.

This is a weaker version of the '∃' elimination rule in the natural deduction system. The version with a non-freeness condition is exia-P6 746.

Hypothesis

Ref	Expression
exiav-P5.1	⊢ (𝜑 → 𝜓)

Assertion

Ref	Expression
exiav-P5	⊢ (∃𝑥𝜑 → 𝜓)

Distinct variable group:   𝜓,𝑥

Proof of Theorem exiav-P5

Step	Hyp	Ref	Expression
1		exiav-P5.1	. . 3 ⊢ (𝜑 → 𝜓)
2	1	alloverimex-P5.RC.GEN 603	. 2 ⊢ (∃𝑥𝜑 → ∃𝑥𝜓)
3		axL5ex-P5 613	. 2 ⊢ (∃𝑥𝜓 → 𝜓)
4	2, 3	syl-P3.24.RC 260	1 ⊢ (∃𝑥𝜑 → 𝜓)

Syntax hints:   → 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

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

This theorem is referenced by:  exiav-P5.SH  616  exipsub-P6  720  spliteq-P6-L1  775  splitelof-P6-L1  777