Theorem 2th 172

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 124	1 ⊢ (𝜑 ↔ 𝜓)

Syntax hints:   ↔ wb 103

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia2 105  ax-ia3 106

This theorem depends on definitions:  df-bi 115

This theorem is referenced by:  trujust  1286  dftru2  1292  bitru  1296  vjust  2602  pwv  3600  int0  3650  0iin  3736  snnex  4199  ruv  4293  fo1st  5804  fo2nd  5805  eqer  6161  ener  6282  rexfiuz  9875  bdth  10622