Theorem dfbi-P3.47 358

Description: Alternate Definition of '↔'. †

The Chapter 2 definition of '↔' comes immediately from combining this definition with the previous definition of '∧'. That definition really isn't useful enough to state though.

Assertion

Ref	Expression
dfbi-P3.47	⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)))

Proof of Theorem dfbi-P3.47

Step	Hyp	Ref	Expression
1		ndbief-P3.14.CL 249	. . 3 ⊢ ((𝜑 ↔ 𝜓) → (𝜑 → 𝜓))
2		ndbier-P3.15.CL 250	. . 3 ⊢ ((𝜑 ↔ 𝜓) → (𝜓 → 𝜑))
3	1, 2	ndandi-P3.7 172	. 2 ⊢ ((𝜑 ↔ 𝜓) → ((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)))
4		ndbii-P3.13.CL 248	. 2 ⊢ (((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)) → (𝜑 ↔ 𝜓))
5	3, 4	rcp-NDBII0 239	1 ⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)))

Syntax hints:   → wff-imp 10   ↔ wff-bi 104   ∧ wff-and 132

This theorem was proved from axioms:  ax-L1 11  ax-L2 12  ax-L3 13  ax-MP 14

This theorem depends on definitions:  df-bi-D2.1 107  df-and-D2.2 133  df-true-D2.4 155

This theorem is referenced by:  oroverbi-P4.28-L3  468  oroverbi-P4.28b  469  imoverbi-P4.30-L2  478  imoverbi-P4.30b  479  biasandor-P4.34a  491  nfrbi-P6  691  nfrbid-P6  818