Theorem pm5.54 1035

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

Assertion

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

Proof of Theorem pm5.54

Step	Hyp	Ref	Expression
1		iba 537	. . . . 5 ⊢ (𝜓 → (𝜑 ↔ (𝜑 ∧ 𝜓)))
2	1	bicomd 226	. . . 4 ⊢ (𝜓 → ((𝜑 ∧ 𝜓) ↔ 𝜑))
3	2	adantl 487	. . 3 ⊢ ((𝜑 ∧ 𝜓) → ((𝜑 ∧ 𝜓) ↔ 𝜑))
4	3, 2	pm5.21ni 380	. 2 ⊢ (¬ ((𝜑 ∧ 𝜓) ↔ 𝜑) → ((𝜑 ∧ 𝜓) ↔ 𝜓))
5	4	orri 876	1 ⊢ (((𝜑 ∧ 𝜓) ↔ 𝜑) ∨ ((𝜑 ∧ 𝜓) ↔ 𝜓))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   ∨ wo 861

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-or 862

This theorem is used by: (None)