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Mirrors > Home > ILE Home > Th. List > tbt | GIF version |
Description: A wff is equivalent to its equivalence with truth. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Andrew Salmon, 13-May-2011.) |
Ref | Expression |
---|---|
tbt.1 | ⊢ 𝜑 |
Ref | Expression |
---|---|
tbt | ⊢ (𝜓 ↔ (𝜓 ↔ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tbt.1 | . 2 ⊢ 𝜑 | |
2 | ibibr 246 | . . 3 ⊢ ((𝜑 → 𝜓) ↔ (𝜑 → (𝜓 ↔ 𝜑))) | |
3 | 2 | pm5.74ri 181 | . 2 ⊢ (𝜑 → (𝜓 ↔ (𝜓 ↔ 𝜑))) |
4 | 1, 3 | ax-mp 5 | 1 ⊢ (𝜓 ↔ (𝜓 ↔ 𝜑)) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 105 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 |
This theorem depends on definitions: df-bi 117 |
This theorem is referenced by: tbtru 1363 exists1 2122 reu6 2926 eqv 3442 vnex 4131 bj-vprc 14297 |
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