Theorem dfbi2 609

Description: A theorem similar to the standard definition of the biconditional. Definition of [Margaris] p. 49. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
dfbi2	⊢ ((φ ↔ ψ) ↔ ((φ → ψ) ∧ (ψ → φ)))

Proof of Theorem dfbi2

Step	Hyp	Ref	Expression
1		dfbi1 184	. 2 ⊢ ((φ ↔ ψ) ↔ ¬ ((φ → ψ) → ¬ (ψ → φ)))
2		df-an 360	. 2 ⊢ (((φ → ψ) ∧ (ψ → φ)) ↔ ¬ ((φ → ψ) → ¬ (ψ → φ)))
3	1, 2	bitr4i 243	1 ⊢ ((φ ↔ ψ) ↔ ((φ → ψ) ∧ (ψ → φ)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358

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 177  df-an 360

This theorem is referenced by:  dfbi  610  pm4.71  611  pm5.17  858  xor  861  albiim  1611  nfbid  1832  nfbiOLD  1835  sbbi  2071  cleqh  2450  ralbiim  2752  reu8  3033  sseq2  3294  funeq  5128  fun11  5160  dffo3  5423