Theorem bitru 1579

Description: A theorem is equivalent to truth. (Contributed by Mario Carneiro, 9-May-2015.)

Hypothesis

Ref	Expression
bitru.1	⊢ 𝜑

Assertion

Ref	Expression
bitru	⊢ (𝜑 ↔ ⊤)

Proof of Theorem bitru

Step	Hyp	Ref	Expression
1		bitru.1	. 2 ⊢ 𝜑
2		tru 1574	. 2 ⊢ ⊤
3	1, 2	2th 267	1 ⊢ (𝜑 ↔ ⊤)

This proof depends on syntax axioms:   ↔ wb 209  ⊤wtru 1571

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

This proof depends on definitions:  df-bi 210  df-tru 1573

This theorem is used by:  truimtru  1593  falimtru  1595  falimfal  1596  notfal  1598  trubitru  1599  falbifal  1602  truorfal  1608  falortru  1609  exists1  2691  dfv2  3461  0frgp  19880  tgcgr4  28837  wl-2mintru1  38177  astbstanbst  47687  atnaiana  47701  dandysum2p2e4  47776