Theorem pm5.1 501

Description: Two propositions are equivalent if they are both true. Theorem *5.1 of [WhiteheadRussell] p. 123.

Assertion

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

Proof of Theorem pm5.1

Step	Hyp	Ref	Expression
1		ibib 448	. . 3 ⊢ ((φ → ψ) ↔ (φ → (φ ↔ ψ)))
2	1	pm5.74ri 445	. 2 ⊢ (φ → (ψ ↔ (φ ↔ ψ)))
3	2	biimpa 324	1 ⊢ ((φ ∧ ψ) → (φ ↔ ψ))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196

This theorem is referenced by:  pm5.21 502  elimant 505  sbc2or 1454  ssconb 1598  ralidm 1774  raaan 1775  eceqopreq 3249  sucdom 3648  zltp1let 4597  mdsym 5784

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6

This theorem depends on definitions:  df-bi 128  df-an 198

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