Metamath Proof Explorer


Theorem iotaeq

Description: Equality theorem for descriptions. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by Andrew Salmon, 30-Jun-2011) (New usage is discouraged.)

Ref Expression
Assertion iotaeq xx=yιx|φ=ιy|φ

Proof

Step Hyp Ref Expression
1 drsb1 xx=yzxφzyφ
2 df-clab zx|φzxφ
3 df-clab zy|φzyφ
4 1 2 3 3bitr4g xx=yzx|φzy|φ
5 4 eqrdv xx=yx|φ=y|φ
6 5 eqeq1d xx=yx|φ=zy|φ=z
7 6 abbidv xx=yz|x|φ=z=z|y|φ=z
8 7 unieqd xx=yz|x|φ=z=z|y|φ=z
9 df-iota ιx|φ=z|x|φ=z
10 df-iota ιy|φ=z|y|φ=z
11 8 9 10 3eqtr4g xx=yιx|φ=ιy|φ