Theorem ecase 1034

Description: Inference for elimination by cases. (Contributed by NM, 13-Jul-2005.)

Hypotheses

Ref	Expression
ecase.1	⊢ (¬ 𝜑 → 𝜒)
ecase.2	⊢ (¬ 𝜓 → 𝜒)
ecase.3	⊢ ((𝜑 ∧ 𝜓) → 𝜒)

Assertion

Ref	Expression
ecase	⊢ 𝜒

Proof of Theorem ecase

Step	Hyp	Ref	Expression
1		ecase.3	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
2	1	ex 412	. 2 ⊢ (𝜑 → (𝜓 → 𝜒))
3		ecase.1	. 2 ⊢ (¬ 𝜑 → 𝜒)
4		ecase.2	. 2 ⊢ (¬ 𝜓 → 𝜒)
5	2, 3, 4	pm2.61nii 184	1 ⊢ 𝜒

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 207  df-an 396

This theorem is referenced by:  sbccomlem  3869  prfi  9363  hashprb  14436  txindislem  23641  iswwlksnon  29873  iswspthsnon  29876  1to3vfriswmgr  30299