Theorem 2th 173

Description: Two truths are equivalent. (Contributed by NM, 18-Aug-1993.)

Hypotheses

Ref	Expression
2th.1	⊢ 𝜑
2th.2	⊢ 𝜓

Assertion

Ref	Expression
2th	⊢ (𝜑 ↔ 𝜓)

Proof of Theorem 2th

Step	Hyp	Ref	Expression
1		2th.2	. . 3 ⊢ 𝜓
2	1	a1i 9	. 2 ⊢ (𝜑 → 𝜓)
3		2th.1	. . 3 ⊢ 𝜑
4	3	a1i 9	. 2 ⊢ (𝜓 → 𝜑)
5	2, 4	impbii 125	1 ⊢ (𝜑 ↔ 𝜓)

Syntax hints:   ↔ wb 104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia2 106  ax-ia3 107

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  trujust  1350  dftru2  1356  bitru  1360  vjust  2731  pwv  3793  int0  3843  0iin  3929  snnex  4431  ruv  4532  fo1st  6134  fo2nd  6135  eqer  6543  ener  6755  rexfiuz  10946  bdth  13831