Theorem 2th 174

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 126	1 ⊢ (𝜑 ↔ 𝜓)

Syntax hints:   ↔ wb 105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia2 107  ax-ia3 108

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  trujust  1399  dftru2  1405  bitru  1409  vjust  2803  pwv  3892  int0  3942  0iin  4029  snnex  4545  ruv  4648  fo1st  6319  fo2nd  6320  eqer  6733  ener  6952  rexfiuz  11549  bdth  16426