Theorem pm5.1 830

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		pm5.501 330	. 2 ⊢ (φ → (ψ ↔ (φ ↔ ψ)))
2	1	biimpa 470	1 ⊢ ((φ ∧ ψ) → (φ ↔ ψ))

This proof depends on syntax axioms:   → wi 4   ↔ wb 176   ∧ wa 358

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 177  df-an 360

This theorem is used by:  pm5.35  869  ssconb  3400  raaan  3658  raaanv  3659