Theorem pm4.71r 482

Description: Implication in terms of biconditional and conjunction. Theorem *4.71 of [WhiteheadRussell] p. 120 (with conjunct reversed).

Assertion

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

Proof of Theorem pm4.71r

Step	Hyp	Ref	Expression
1		pm4.71 481	. 2 ⊢ ((φ → ψ) ↔ (φ ↔ (φ ∧ ψ)))
2		ancom 333	. . 3 ⊢ ((φ ∧ ψ) ↔ (ψ ∧ φ))
3	2	bibi2i 460	. 2 ⊢ ((φ ↔ (φ ∧ ψ)) ↔ (φ ↔ (ψ ∧ φ)))
4	1, 3	bitr 151	1 ⊢ ((φ → ψ) ↔ (φ ↔ (ψ ∧ φ)))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196

This theorem is referenced by:  pm4.71ri 484  bimsc1 557  reuhyp 1581  ordsucun 2333  iss 2599  fcoi1 2765  feu 2767  fnopabfv 2858  fniunfv 2860  shselt 5280

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6

This theorem depends on definitions:  df-bi 128  df-an 198

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