Theorem pm4.71 557

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 482	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜑)
2	1	biantru 529	. 2 ⊢ ((𝜑 → (𝜑 ∧ 𝜓)) ↔ ((𝜑 → (𝜑 ∧ 𝜓)) ∧ ((𝜑 ∧ 𝜓) → 𝜑)))
3		anclb 545	. 2 ⊢ ((𝜑 → 𝜓) ↔ (𝜑 → (𝜑 ∧ 𝜓)))
4		dfbi2 474	. 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 206   ∧ wa 395

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 207  df-an 396

This theorem is referenced by:  pm4.71r  558  pm4.71i  559  pm4.71d  561  bigolden  1028  rabid2f  3467  rabid2im  3468  rabid2OLD  3470  disj3  4453  dmopab3  5929  rnopab3  5966  mptfnf  6702  wl-ifp4impr  37469  nanorxor  44329