Theorem pm5.1 836

Description: Two propositions are equivalent if they are both true. Theorem *5.1 of [WhiteheadRussell] p. 123. (Contributed by NM, 21-May-1994.)

Assertion

Ref	Expression
pm5.1	⊢ ((𝜑 ∧ 𝜓) → (𝜑 ↔ 𝜓))

Proof of Theorem pm5.1

Step	Hyp	Ref	Expression
1		pm5.501 369	. 2 ⊢ (𝜑 → (𝜓 ↔ (𝜑 ↔ 𝜓)))
2	1	biimpa 482	1 ⊢ ((𝜑 ∧ 𝜓) → (𝜑 ↔ 𝜓))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401

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

This theorem is used by:  pm5.35  838  abab  840  impimprbi  842  ssconb  4099  raaan  4484  raaanv  4485  raaan2  4488  suppimacnvss  8178  mdsymi  32800  ply1degltel  33915  ply1degleel  33916  tsbi1  38823  abnotbtaxb  47693  elprneb  47807