Theorem rp-fakeoranass

Description: A special case where a mixture of or and and appears to conform to a mixed associative law. (Contributed by RP, 29-Feb-2020)

	Ref	Expression
Assertion	rp-fakeoranass	⊢ ( ( 𝜑 → 𝜒 ) ↔ ( ( ( 𝜑 ∨ 𝜓 ) ∧ 𝜒 ) ↔ ( 𝜑 ∨ ( 𝜓 ∧ 𝜒 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		rp-fakeanorass	⊢ ( ( 𝜑 → 𝜒 ) ↔ ( ( ( 𝜒 ∧ 𝜓 ) ∨ 𝜑 ) ↔ ( 𝜒 ∧ ( 𝜓 ∨ 𝜑 ) ) ) )
2		bicom	⊢ ( ( ( ( 𝜒 ∧ 𝜓 ) ∨ 𝜑 ) ↔ ( 𝜒 ∧ ( 𝜓 ∨ 𝜑 ) ) ) ↔ ( ( 𝜒 ∧ ( 𝜓 ∨ 𝜑 ) ) ↔ ( ( 𝜒 ∧ 𝜓 ) ∨ 𝜑 ) ) )
3		orcom	⊢ ( ( 𝜓 ∨ 𝜑 ) ↔ ( 𝜑 ∨ 𝜓 ) )
4	3	anbi1ci	⊢ ( ( 𝜒 ∧ ( 𝜓 ∨ 𝜑 ) ) ↔ ( ( 𝜑 ∨ 𝜓 ) ∧ 𝜒 ) )
5		orcom	⊢ ( ( ( 𝜒 ∧ 𝜓 ) ∨ 𝜑 ) ↔ ( 𝜑 ∨ ( 𝜒 ∧ 𝜓 ) ) )
6		ancom	⊢ ( ( 𝜒 ∧ 𝜓 ) ↔ ( 𝜓 ∧ 𝜒 ) )
7	6	orbi2i	⊢ ( ( 𝜑 ∨ ( 𝜒 ∧ 𝜓 ) ) ↔ ( 𝜑 ∨ ( 𝜓 ∧ 𝜒 ) ) )
8	5 7	bitri	⊢ ( ( ( 𝜒 ∧ 𝜓 ) ∨ 𝜑 ) ↔ ( 𝜑 ∨ ( 𝜓 ∧ 𝜒 ) ) )
9	4 8	bibi12i	⊢ ( ( ( 𝜒 ∧ ( 𝜓 ∨ 𝜑 ) ) ↔ ( ( 𝜒 ∧ 𝜓 ) ∨ 𝜑 ) ) ↔ ( ( ( 𝜑 ∨ 𝜓 ) ∧ 𝜒 ) ↔ ( 𝜑 ∨ ( 𝜓 ∧ 𝜒 ) ) ) )
10	2 9	bitri	⊢ ( ( ( ( 𝜒 ∧ 𝜓 ) ∨ 𝜑 ) ↔ ( 𝜒 ∧ ( 𝜓 ∨ 𝜑 ) ) ) ↔ ( ( ( 𝜑 ∨ 𝜓 ) ∧ 𝜒 ) ↔ ( 𝜑 ∨ ( 𝜓 ∧ 𝜒 ) ) ) )
11	1 10	bitri	⊢ ( ( 𝜑 → 𝜒 ) ↔ ( ( ( 𝜑 ∨ 𝜓 ) ∧ 𝜒 ) ↔ ( 𝜑 ∨ ( 𝜓 ∧ 𝜒 ) ) ) )