Theorem abab 835

Description: Introduce one conjunct as equivalent to the other. "abab" stands for "and, biconditional, and, biconditional". (Contributed by Wolf Lammen, 4-Jun-2026.)

Assertion

Ref	Expression
abab	⊢ ((𝜑 ∧ 𝜓) ↔ (𝜑 ∧ (𝜑 ↔ 𝜓)))

Proof of Theorem abab

Step	Hyp	Ref	Expression
1		simpl 485	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜑)
2		pm5.1 831	. . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝜑 ↔ 𝜓))
3	1, 2	jca 518	. 2 ⊢ ((𝜑 ∧ 𝜓) → (𝜑 ∧ (𝜑 ↔ 𝜓)))
4		biimp 217	. . . 4 ⊢ ((𝜑 ↔ 𝜓) → (𝜑 → 𝜓))
5	4	anim2i 625	. . 3 ⊢ ((𝜑 ∧ (𝜑 ↔ 𝜓)) → (𝜑 ∧ (𝜑 → 𝜓)))
6		abai 834	. . 3 ⊢ ((𝜑 ∧ 𝜓) ↔ (𝜑 ∧ (𝜑 → 𝜓)))
7	5, 6	sylibr 236	. 2 ⊢ ((𝜑 ∧ (𝜑 ↔ 𝜓)) → (𝜑 ∧ 𝜓))
8	3, 7	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:  just2-df  2081  just3-df  2082