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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) Remove dependency on ax-10 , ax-11 , ax-12 . (Revised by SN, 23-Nov-2024)

Ref Expression
Assertion iotaval ⊢ ∀ x φ ↔ x = y → ι x | φ = y

Proof

Step Hyp Ref Expression
1 abbi ⊢ ∀ x φ ↔ x = y → x | φ = x | x = y
2 df-sn ⊢ y = x | x = y
3 1 2 eqtr4di ⊢ ∀ x φ ↔ x = y → x | φ = y
4 iotaval2 ⊢ x | φ = y → ι x | φ = y
5 3 4 syl ⊢ ∀ x φ ↔ x = y → ι x | φ = y