Theorem ecased 1238

Description: Deduction form of disjunctive syllogism. (Contributed by Jim Kingdon, 9-Dec-2017.)

Hypotheses

Ref	Expression
ecased.1	⊢ (φ → ¬ χ)
ecased.2	⊢ (φ → (ψ ∨ χ))

Assertion

Ref	Expression
ecased	⊢ (φ → ψ)

Proof of Theorem ecased

Step	Hyp	Ref	Expression
1		ecased.1	. . 3 ⊢ (φ → ¬ χ)
2		ecased.2	. . 3 ⊢ (φ → (ψ ∨ χ))
3	1, 2	jca 290	. 2 ⊢ (φ → (¬ χ ∧ (ψ ∨ χ)))
4		orel2 644	. . 3 ⊢ (¬ χ → ((ψ ∨ χ) → ψ))
5	4	imp 115	. 2 ⊢ ((¬ χ ∧ (ψ ∨ χ)) → ψ)
6	3, 5	syl 14	1 ⊢ (φ → ψ)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ∨ wo 628

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in2 545  ax-io 629

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  ecase23d  1239  preleq  4233  ordsuc  4241  sotri3  4666  addnqprlemfl  6539  addnqprlemfu  6540  addcanprleml  6587  addcanprlemu  6588  cauappcvgprlemladdru  6627  cauappcvgprlemladdrl  6628  ltletr  6884  apreap  7351  ltleap  7393  uzm1  8259  xrltletr  8473