Theorem pm4.71 393

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 109	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜑)
2	1	biantru 302	. 2 ⊢ ((𝜑 → (𝜑 ∧ 𝜓)) ↔ ((𝜑 → (𝜑 ∧ 𝜓)) ∧ ((𝜑 ∧ 𝜓) → 𝜑)))
3		anclb 319	. 2 ⊢ ((𝜑 → 𝜓) ↔ (𝜑 → (𝜑 ∧ 𝜓)))
4		dfbi2 392	. 2 ⊢ ((𝜑 ↔ (𝜑 ∧ 𝜓)) ↔ ((𝜑 → (𝜑 ∧ 𝜓)) ∧ ((𝜑 ∧ 𝜓) → 𝜑)))
5	2, 3, 4	3bitr4i 212	1 ⊢ ((𝜑 → 𝜓) ↔ (𝜑 ↔ (𝜑 ∧ 𝜓)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ↔ wb 105

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This proof depends on definitions:  df-bi 117

This theorem is used by:  pm4.71r  394  pm4.71i  395  pm4.71d  397  bigolden  968  pm5.75  975  exintrbi  1686  rabid2  2729  ssalel  3235  disj3  3577  dmopab3  4994  dfalseu2  17177