Theorem pm4.71 558

Description: Implication in terms of biconditional and conjunction. Theorem *4.71 of [WhiteheadRussell] p. 120. (Contributed by NM, 21-Jun-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 483	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜑)
2	1	biantru 530	. 2 ⊢ ((𝜑 → (𝜑 ∧ 𝜓)) ↔ ((𝜑 → (𝜑 ∧ 𝜓)) ∧ ((𝜑 ∧ 𝜓) → 𝜑)))
3		anclb 546	. 2 ⊢ ((𝜑 → 𝜓) ↔ (𝜑 → (𝜑 ∧ 𝜓)))
4		dfbi2 475	. 2 ⊢ ((𝜑 ↔ (𝜑 ∧ 𝜓)) ↔ ((𝜑 → (𝜑 ∧ 𝜓)) ∧ ((𝜑 ∧ 𝜓) → 𝜑)))
5	2, 3, 4	3bitr4i 303	1 ⊢ ((𝜑 → 𝜓) ↔ (𝜑 ↔ (𝜑 ∧ 𝜓)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396

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 206  df-an 397

This theorem is referenced by:  pm4.71r  559  pm4.71i  560  pm4.71d  562  bigolden  1024  rabid2f  3313  rabid2OLD  3315  dfss2  3907  dfss2OLD  3908  disj3  4387  dmopab3  5828  mptfnf  6568  wl-ifp4impr  35638  nanorxor  41923