Theorem 3impcombi 42326

Description: A 1-hypothesis propositional calculus deduction. (Contributed by Alan Sare, 25-Sep-2017.)

Hypothesis

Ref	Expression
3impcombi.1	⊢ ((𝜑 ∧ 𝜓 ∧ 𝜑) → (𝜒 ↔ 𝜃))

Assertion

Ref	Expression
3impcombi	⊢ ((𝜓 ∧ 𝜑 ∧ 𝜒) → 𝜃)

Proof of Theorem 3impcombi

Step	Hyp	Ref	Expression
1		3impcombi.1	. . . . 5 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜑) → (𝜒 ↔ 𝜃))
2	1	biimpd 228	. . . 4 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜑) → (𝜒 → 𝜃))
3	2	3anidm13 1418	. . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝜒 → 𝜃))
4	3	ancoms 458	. 2 ⊢ ((𝜓 ∧ 𝜑) → (𝜒 → 𝜃))
5	4	3impia 1115	1 ⊢ ((𝜓 ∧ 𝜑 ∧ 𝜒) → 𝜃)

Syntax hints:   → wi 4   ↔ wb 205   ∧ w3a 1085

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 206  df-an 396  df-3an 1087

This theorem is referenced by:  isosctrlem1ALT  42443