Theorem 2th 230

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 10	. 2 ⊢ (φ → ψ)
3		2th.1	. . 3 ⊢ φ
4	3	a1i 10	. 2 ⊢ (ψ → φ)
5	2, 4	impbii 180	1 ⊢ (φ ↔ ψ)

This proof depends on syntax axioms:   ↔ wb 176

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 177

This theorem is used by:  2false  339  trujust  1318  bitru  1326  exiftruOLD  1658  pm11.07  2115  vjust  2861  dfnul2  3553  dfnul3  3554  rab0  3572  pwv  3887  int0  3941  0iin  4025