Theorem pm4.71 481

Description: Implication in terms of biconditional and conjunction. Theorem *4.71 of [WhiteheadRussell] p. 120.

Assertion

Ref	Expression
pm4.71	⊢ ((φ → ψ) ↔ (φ ↔ (φ ∧ ψ)))

Proof of Theorem pm4.71

Step	Hyp	Ref	Expression
1		ancl 242	. . 3 ⊢ ((φ → ψ) → (φ → (φ ∧ ψ)))
2		pm3.26 256	. . . 4 ⊢ ((φ ∧ ψ) → φ)
3	2	a1i 7	. . 3 ⊢ ((φ → ψ) → ((φ ∧ ψ) → φ))
4	1, 3	impbid 397	. 2 ⊢ ((φ → ψ) → (φ ↔ (φ ∧ ψ)))
5		bi1 130	. . 3 ⊢ ((φ ↔ (φ ∧ ψ)) → (φ → (φ ∧ ψ)))
6		pm3.27 260	. . 3 ⊢ ((φ ∧ ψ) → ψ)
7	5, 6	syl6 23	. 2 ⊢ ((φ ↔ (φ ∧ ψ)) → (φ → ψ))
8	4, 7	impbi 139	1 ⊢ ((φ → ψ) ↔ (φ ↔ (φ ∧ ψ)))

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

This theorem is referenced by:  pm4.71r 482  pm4.71i 483  bigolden 513  rabid2 1309  dfss2 1497  disj3 1736  moabex 1868  dmopab2 2541  fcoi2 2766  fcnvres 2768  pw2en 3348

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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