Theorem bitru 1578

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 1573	. 2 ⊢ ⊤
3	1, 2	2th 267	1 ⊢ (𝜑 ↔ ⊤)

This proof depends on syntax axioms:   ↔ wb 209  ⊤wtru 1570

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 1572

This theorem is used by:  truimtru  1592  falimtru  1594  falimfal  1595  notfal  1597  trubitru  1598  falbifal  1601  truorfal  1607  falortru  1608  exists1  2685  dfv2  3453  0frgp  19999  tgcgr4  29000  wl-2mintru1  38507  astbstanbst  48073  atnaiana  48087  dandysum2p2e4  48162