Theorem pm4.71 566

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 487	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜑)
2	1	biantru 538	. 2 ⊢ ((𝜑 → (𝜑 ∧ 𝜓)) ↔ ((𝜑 → (𝜑 ∧ 𝜓)) ∧ ((𝜑 ∧ 𝜓) → 𝜑)))
3		anclb 554	. 2 ⊢ ((𝜑 → 𝜓) ↔ (𝜑 → (𝜑 ∧ 𝜓)))
4		dfbi2 479	. 2 ⊢ ((𝜑 ↔ (𝜑 ∧ 𝜓)) ↔ ((𝜑 → (𝜑 ∧ 𝜓)) ∧ ((𝜑 ∧ 𝜓) → 𝜑)))
5	2, 3, 4	3bitr4i 306	1 ⊢ ((𝜑 → 𝜓) ↔ (𝜑 ↔ (𝜑 ∧ 𝜓)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 401

This theorem is used by:  pm4.71r  567  pm4.71i  568  pm4.71d  570  bigolden  1043  rabid2f  3446  rabid2im  3447  disj3  4413  dmopab3  5908  rnopab3  5945  mptfnf  6670  wl-ifp4impr  38141  nanorxor  45043  dfalseu2  50642