Theorem bitru 1547

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 1542	. 2 ⊢ ⊤
3	1, 2	2th 267	1 ⊢ (𝜑 ↔ ⊤)

Syntax hints:   ↔ wb 209  ⊤wtru 1539

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 210  df-tru 1541

This theorem is referenced by:  truimtru  1561  falimtru  1563  falimfal  1564  notfal  1566  trubitru  1567  falbifal  1570  truorfal  1576  falortru  1577  exists1  2723  0frgp  18897  tgcgr4  26325  wl-2mintru1  34907  astbstanbst  43502  atnaiana  43516  dandysum2p2e4  43591