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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 ⊢ ∀ x x = y → ι x | φ = ι y | φ

Proof

Step Hyp Ref Expression
1 drsb1 ⊢ ∀ x x = y → z x φ ↔ z y φ
2 df-clab ⊢ z ∈ x | φ ↔ z x φ
3 df-clab ⊢ z ∈ y | φ ↔ z y φ
4 1 2 3 3bitr4g ⊢ ∀ x x = y → z ∈ x | φ ↔ z ∈ y | φ
5 4 eqrdv ⊢ ∀ x x = y → x | φ = y | φ
6 5 eqeq1d ⊢ ∀ x x = y → x | φ = z ↔ y | φ = z
7 6 abbidv ⊢ ∀ x x = y → z | x | φ = z = z | y | φ = z
8 7 unieqd ⊢ ∀ x x = y → ⋃ z | x | φ = z = ⋃ z | y | φ = z
9 df-iota ⊢ ι x | φ = ⋃ z | x | φ = z
10 df-iota ⊢ ι y | φ = ⋃ z | y | φ = z
11 8 9 10 3eqtr4g ⊢ ∀ x x = y → ι x | φ = ι y | φ