Theorem 3impcombi 45758

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 232	. . . 4 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜑) → (𝜒 → 𝜃))
3	2	3anidm13 1447	. . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝜒 → 𝜃))
4	3	ancoms 464	. 2 ⊢ ((𝜓 ∧ 𝜑) → (𝜒 → 𝜃))
5	4	3impia 1135	1 ⊢ ((𝜓 ∧ 𝜑 ∧ 𝜒) → 𝜃)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ w3a 1103

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  df-3an 1105

This theorem is used by:  isosctrlem1ALT  45875