Metamath Proof Explorer


Theorem log11d

Description: The natural logarithm is one-to-one. (Contributed by SN, 25-Apr-2025)

Ref Expression
Hypotheses log11d.a φ A
log11d.b φ B
log11d.1 φ A 0
log11d.2 φ B 0
Assertion log11d φ log A = log B A = B

Proof

Step Hyp Ref Expression
1 log11d.a φ A
2 log11d.b φ B
3 log11d.1 φ A 0
4 log11d.2 φ B 0
5 fveq2 log A = log B e log A = e log B
6 eflog A A 0 e log A = A
7 1 3 6 syl2anc φ e log A = A
8 eflog B B 0 e log B = B
9 2 4 8 syl2anc φ e log B = B
10 7 9 eqeq12d φ e log A = e 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