Theorem tbt 541

Description: A wff is equivalent to its equivalence with truth.

Hypothesis

Ref	Expression
tbt.1	⊢ φ

Assertion

Ref	Expression
tbt	⊢ (ψ ↔ (ψ ↔ φ))

Proof of Theorem tbt

Step	Hyp	Ref	Expression
1		tbt.1	. . . . 5 ⊢ φ
2	1	a1i 7	. . . 4 ⊢ (ψ → φ)
3	2	a1d 14	. . 3 ⊢ (ψ → (ψ → φ))
4		ax-1 3	. . 3 ⊢ (ψ → (φ → ψ))
5	3, 4	impbid 397	. 2 ⊢ (ψ → (ψ ↔ φ))
6		bi2 131	. . 3 ⊢ ((ψ ↔ φ) → (φ → ψ))
7	1, 6	mpi 44	. 2 ⊢ ((ψ ↔ φ) → ψ)
8	5, 7	impbi 139	1 ⊢ (ψ ↔ (ψ ↔ φ))

Syntax hints:   ↔ wb 127

This theorem is referenced by:  exists1 1072  nvelv 1483

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

This theorem depends on definitions:  df-bi 128  df-an 198

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