Theorem tbt 719

Description: A wff is equivalent to its equivalence with truth. (The proof was shortened by Juha Arpiainen, 19-Jan-2006.)

Hypothesis

Ref	Expression
tbt.1	⊢ φ

Assertion

Ref	Expression
tbt	⊢ (ψ ↔ (ψ ↔ φ))

Proof of Theorem tbt

Step	Hyp	Ref	Expression
1		tbt.1	. . 3 ⊢ φ
2		pm5.501 594	. . 3 ⊢ (φ → (ψ ↔ (φ ↔ ψ)))
3	1, 2	ax-mp 7	. 2 ⊢ (ψ ↔ (φ ↔ ψ))
4		bicom 519	. 2 ⊢ ((φ ↔ ψ) ↔ (ψ ↔ φ))
5	3, 4	bitr 173	1 ⊢ (ψ ↔ (ψ ↔ φ))

Syntax hints:   ↔ wb 146

This theorem is referenced by:  exists1 1456  reu3 1928  eqv 2292  nvelv 2709  asymref2 3436

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7

This theorem depends on definitions:  df-bi 147  df-an 225

Copyright terms: Public domain