Theorem contrd 38779

Description: A proof by contradiction, in deduction form. (Contributed by Giovanni Mascellani, 19-Mar-2018.)

Hypotheses

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

Assertion

Ref	Expression
contrd	⊢ (𝜑 → 𝜓)

Proof of Theorem contrd

Step	Hyp	Ref	Expression
1		contrd.1	. . 3 ⊢ (𝜑 → (¬ 𝜓 → 𝜒))
2		contrd.2	. . 3 ⊢ (𝜑 → (¬ 𝜓 → ¬ 𝜒))
3	1, 2	jcad 522	. 2 ⊢ (𝜑 → (¬ 𝜓 → (𝜒 ∧ ¬ 𝜒)))
4		pm2.24 125	. . . . 5 ⊢ (𝜒 → (¬ 𝜒 → 𝜓))
5	4	imp 412	. . . 4 ⊢ ((𝜒 ∧ ¬ 𝜒) → 𝜓)
6	5	imim2i 17	. . 3 ⊢ ((¬ 𝜓 → (𝜒 ∧ ¬ 𝜒)) → (¬ 𝜓 → 𝜓))
7	6	pm2.18d 128	. 2 ⊢ ((¬ 𝜓 → (𝜒 ∧ ¬ 𝜒)) → 𝜓)
8	3, 7	syl 18	1 ⊢ (𝜑 → 𝜓)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402

This theorem is used by:  mpobi123f  38844  mptbi12f  38848  ac6s6  38854