Metamath Proof Explorer
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)
|
|
Ref |
Expression |
|
Hypotheses |
bnj1541.1 |
⊢ ( 𝜑 ↔ ( 𝜓 ∧ 𝐴 ≠ 𝐵 ) ) |
|
|
bnj1541.2 |
⊢ ¬ 𝜑 |
|
Assertion |
bnj1541 |
⊢ ( 𝜓 → 𝐴 = 𝐵 ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
bnj1541.1 |
⊢ ( 𝜑 ↔ ( 𝜓 ∧ 𝐴 ≠ 𝐵 ) ) |
2 |
|
bnj1541.2 |
⊢ ¬ 𝜑 |
3 |
2 1
|
mtbi |
⊢ ¬ ( 𝜓 ∧ 𝐴 ≠ 𝐵 ) |
4 |
3
|
imnani |
⊢ ( 𝜓 → ¬ 𝐴 ≠ 𝐵 ) |
5 |
|
nne |
⊢ ( ¬ 𝐴 ≠ 𝐵 ↔ 𝐴 = 𝐵 ) |
6 |
4 5
|
sylib |
⊢ ( 𝜓 → 𝐴 = 𝐵 ) |