Theorem pm5.1 823

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		pm5.501 366	. 2 ⊢ (𝜑 → (𝜓 ↔ (𝜑 ↔ 𝜓)))
2	1	biimpa 476	1 ⊢ ((𝜑 ∧ 𝜓) → (𝜑 ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 207  df-an 396

This theorem is referenced by:  pm5.35  825  impimprbi  828  ssconb  4165  raaan  4540  raaanv  4541  raaan2  4544  suppimacnvss  8214  mdsymi  32443  ply1degltel  33580  ply1degleel  33581  tsbi1  38093  abnotbtaxb  46830  elprneb  46944