Theorem exe-P7 955

Description: Simplified '∃' Elimination Law. †

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

Hypotheses

Ref	Expression
exe-P7.1	⊢ Ⅎ𝑥𝛾
exe-P7.2	⊢ Ⅎ𝑥𝜓
exe-P7.3	⊢ (𝛾 → (𝜑 → 𝜓))
exe-P7.4	⊢ (𝛾 → ∃𝑥𝜑)

Assertion

Ref	Expression
exe-P7	⊢ (𝛾 → 𝜓)

Proof of Theorem exe-P7

Step	Hyp	Ref	Expression
1		exe-P7.4	. 2 ⊢ (𝛾 → ∃𝑥𝜑)
2		exe-P7.1	. . 3 ⊢ Ⅎ𝑥𝛾
3		exe-P7.2	. . 3 ⊢ Ⅎ𝑥𝜓
4		exe-P7.3	. . 3 ⊢ (𝛾 → (𝜑 → 𝜓))
5	2, 3, 4	exia-P7 953	. 2 ⊢ (𝛾 → (∃𝑥𝜑 → 𝜓))
6	1, 5	ndime-P3.6 171	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:  allnegex-P7-L1  956  exe-P7r  998  exe-P7r.VR1of2  999  exe-P7r.VR2of2  1000  exe-P7r.VR12of2  1001  exe-P7r.RC  1002  exnegallint-P7  1047  qimeqex-P7-L1  1054  qimeqex-P7-L2  1055