Theorem exe-P7r 998

Description: Simplified '∃' Elimination Law. †

For the original form, using explicit substitution, see ndexe-P7.20 845 and ndexew-P7 867.

This is the restatement of a previously proven result. Do not use in proofs. Use exe-P7 955 instead.

Hypotheses

Ref	Expression
exe-P7r.1	⊢ Ⅎ𝑥𝛾
exe-P7r.2	⊢ Ⅎ𝑥𝜓
exe-P7r.3	⊢ (𝛾 → (𝜑 → 𝜓))
exe-P7r.4	⊢ (𝛾 → ∃𝑥𝜑)

Assertion

Ref	Expression
exe-P7r	⊢ (𝛾 → 𝜓)

Proof of Theorem exe-P7r

Step	Hyp	Ref	Expression
1		exe-P7r.1	. 2 ⊢ Ⅎ𝑥𝛾
2		exe-P7r.2	. 2 ⊢ Ⅎ𝑥𝜓
3		exe-P7r.3	. 2 ⊢ (𝛾 → (𝜑 → 𝜓))
4		exe-P7r.4	. 2 ⊢ (𝛾 → ∃𝑥𝜑)
5	1, 2, 3, 4	exe-P7 955	1 ⊢ (𝛾 → 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: (None)