Theorem pm5.54

Description: Theorem *5.54 of WhiteheadRussell p. 125. (Contributed by NM, 3-Jan-2005) (Proof shortened by Wolf Lammen, 7-Nov-2013)

	Ref	Expression
Assertion	pm5.54	⊢ ( ( ( 𝜑 ∧ 𝜓 ) ↔ 𝜑 ) ∨ ( ( 𝜑 ∧ 𝜓 ) ↔ 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		iba	⊢ ( 𝜓 → ( 𝜑 ↔ ( 𝜑 ∧ 𝜓 ) ) )
2	1	bicomd	⊢ ( 𝜓 → ( ( 𝜑 ∧ 𝜓 ) ↔ 𝜑 ) )
3	2	adantl	⊢ ( ( 𝜑 ∧ 𝜓 ) → ( ( 𝜑 ∧ 𝜓 ) ↔ 𝜑 ) )
4	3 2	pm5.21ni	⊢ ( ¬ ( ( 𝜑 ∧ 𝜓 ) ↔ 𝜑 ) → ( ( 𝜑 ∧ 𝜓 ) ↔ 𝜓 ) )
5	4	orri	⊢ ( ( ( 𝜑 ∧ 𝜓 ) ↔ 𝜑 ) ∨ ( ( 𝜑 ∧ 𝜓 ) ↔ 𝜓 ) )