Theorem pm4.71 611

Description: Implication in terms of biconditional and conjunction. Theorem *4.71 of [WhiteheadRussell] p. 120. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 2-Dec-2012.)

Assertion

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

Proof of Theorem pm4.71

Step	Hyp	Ref	Expression
1		simpl 443	. . 3 ⊢ ((φ ∧ ψ) → φ)
2	1	biantru 491	. 2 ⊢ ((φ → (φ ∧ ψ)) ↔ ((φ → (φ ∧ ψ)) ∧ ((φ ∧ ψ) → φ)))
3		anclb 530	. 2 ⊢ ((φ → ψ) ↔ (φ → (φ ∧ ψ)))
4		dfbi2 609	. 2 ⊢ ((φ ↔ (φ ∧ ψ)) ↔ ((φ → (φ ∧ ψ)) ∧ ((φ ∧ ψ) → φ)))
5	2, 3, 4	3bitr4i 268	1 ⊢ ((φ → ψ) ↔ (φ ↔ (φ ∧ ψ)))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358

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 177  df-an 360

This theorem is referenced by:  pm4.71r  612  pm4.71i  613  pm4.71d  615  bigolden  901  pm5.75  903  exintrbi  1613  rabid2  2789  dfss2  3263  disj3  3596  dmopab3  4918  resopab2  5002  resoprab2  5583