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Metamath Proof Explorer
Theorem tbt
Description: A wff is equivalent to its equivalence with a truth. (Contributed by NM, 18-Aug-1993) (Proof shortened by Andrew Salmon, 13-May-2011)
|
|
Ref |
Expression |
|
Hypothesis |
tbt.1 |
⊢ 𝜑 |
|
Assertion |
tbt |
⊢ ( 𝜓 ↔ ( 𝜓 ↔ 𝜑 ) ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
tbt.1 |
⊢ 𝜑 |
| 2 |
|
ibibr |
⊢ ( ( 𝜑 → 𝜓 ) ↔ ( 𝜑 → ( 𝜓 ↔ 𝜑 ) ) ) |
| 3 |
2
|
pm5.74ri |
⊢ ( 𝜑 → ( 𝜓 ↔ ( 𝜓 ↔ 𝜑 ) ) ) |
| 4 |
1 3
|
ax-mp |
⊢ ( 𝜓 ↔ ( 𝜓 ↔ 𝜑 ) ) |