Metamath Proof Explorer


Theorem iotaval

Description: Theorem 8.19 in Quine p. 57. This theorem is the fundamental property of iota. (Contributed by Andrew Salmon, 11-Jul-2011)

Ref Expression
Assertion iotaval x φ x = y ι x | φ = y

Proof

Step Hyp Ref Expression
1 dfiota2 ι x | φ = z | x φ x = z
2 sbeqalb y V x φ x = y x φ x = z y = z
3 2 elv x φ x = y x φ x = z y = z
4 3 ex x φ x = y x φ x = z y = z
5 equequ2 y = z x = y x = z
6 5 bibi2d y = z φ x = y φ x = z
7 6 biimpd y = z φ x = y φ x = z
8 7 alimdv y = z x φ x = y x φ x = z
9 8 com12 x φ x = y y = z x φ x = z
10 4 9 impbid x φ x = y x φ x = z y = z
11 equcom y = z z = y
12 10 11 syl6bb x φ x = y x φ x = z z = y
13 12 alrimiv x φ x = y z x φ x = z z = y
14 uniabio z x φ x = z z = y z | x φ x = z = y
15 13 14 syl x φ x = y z | x φ x = z = y
16 1 15 syl5eq x φ x = y ι x | φ = y