Step |
Hyp |
Ref |
Expression |
1 |
|
log11d.a |
⊢ ( 𝜑 → 𝐴 ∈ ℂ ) |
2 |
|
log11d.b |
⊢ ( 𝜑 → 𝐵 ∈ ℂ ) |
3 |
|
log11d.1 |
⊢ ( 𝜑 → 𝐴 ≠ 0 ) |
4 |
|
log11d.2 |
⊢ ( 𝜑 → 𝐵 ≠ 0 ) |
5 |
|
fveq2 |
⊢ ( ( log ‘ 𝐴 ) = ( log ‘ 𝐵 ) → ( exp ‘ ( log ‘ 𝐴 ) ) = ( exp ‘ ( log ‘ 𝐵 ) ) ) |
6 |
|
eflog |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( exp ‘ ( log ‘ 𝐴 ) ) = 𝐴 ) |
7 |
1 3 6
|
syl2anc |
⊢ ( 𝜑 → ( exp ‘ ( log ‘ 𝐴 ) ) = 𝐴 ) |
8 |
|
eflog |
⊢ ( ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( exp ‘ ( log ‘ 𝐵 ) ) = 𝐵 ) |
9 |
2 4 8
|
syl2anc |
⊢ ( 𝜑 → ( exp ‘ ( log ‘ 𝐵 ) ) = 𝐵 ) |
10 |
7 9
|
eqeq12d |
⊢ ( 𝜑 → ( ( exp ‘ ( log ‘ 𝐴 ) ) = ( exp ‘ ( log ‘ 𝐵 ) ) ↔ 𝐴 = 𝐵 ) ) |
11 |
5 10
|
imbitrid |
⊢ ( 𝜑 → ( ( log ‘ 𝐴 ) = ( log ‘ 𝐵 ) → 𝐴 = 𝐵 ) ) |
12 |
|
fveq2 |
⊢ ( 𝐴 = 𝐵 → ( log ‘ 𝐴 ) = ( log ‘ 𝐵 ) ) |
13 |
11 12
|
impbid1 |
⊢ ( 𝜑 → ( ( log ‘ 𝐴 ) = ( log ‘ 𝐵 ) ↔ 𝐴 = 𝐵 ) ) |