Theorem bj-animbi 33918

Description: Conjunction in terms of implication and biconditional. Note that the proof is intuitionistic (use of ax-3 8 comes from the unusual definition of the biconditional in set.mm). (Contributed by BJ, 23-Sep-2023.)

Assertion

Ref	Expression
bj-animbi	⊢ ((𝜑 ∧ 𝜓) ↔ (𝜑 ↔ (𝜑 → 𝜓)))

Proof of Theorem bj-animbi

Step	Hyp	Ref	Expression
1		simpl 485	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜑)
2		pm3.4 808	. . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝜑 → 𝜓))
3	1, 2	2thd 267	. 2 ⊢ ((𝜑 ∧ 𝜓) → (𝜑 ↔ (𝜑 → 𝜓)))
4		biimp 217	. . . . 5 ⊢ ((𝜑 ↔ (𝜑 → 𝜓)) → (𝜑 → (𝜑 → 𝜓)))
5	4	pm2.43d 53	. . . 4 ⊢ ((𝜑 ↔ (𝜑 → 𝜓)) → (𝜑 → 𝜓))
6		biimpr 222	. . . 4 ⊢ ((𝜑 ↔ (𝜑 → 𝜓)) → ((𝜑 → 𝜓) → 𝜑))
7	5, 6	mpd 15	. . 3 ⊢ ((𝜑 ↔ (𝜑 → 𝜓)) → 𝜑)
8	7, 5	jcai 519	. 2 ⊢ ((𝜑 ↔ (𝜑 → 𝜓)) → (𝜑 ∧ 𝜓))
9	3, 8	impbii 211	1 ⊢ ((𝜑 ∧ 𝜓) ↔ (𝜑 ↔ (𝜑 → 𝜓)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398

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 209  df-an 399

This theorem is referenced by:  bj-currypara  33919