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Metamath Proof Explorer

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 ) )