Metamath Proof Explorer


Theorem iota5f

Description: A method for computing iota. (Contributed by Scott Fenton, 13-Dec-2017)

Ref Expression
Hypotheses iota5f.1 𝑥 𝜑
iota5f.2 𝑥 𝐴
iota5f.3 ( ( 𝜑𝐴𝑉 ) → ( 𝜓𝑥 = 𝐴 ) )
Assertion iota5f ( ( 𝜑𝐴𝑉 ) → ( ℩ 𝑥 𝜓 ) = 𝐴 )

Proof

Step Hyp Ref Expression
1 iota5f.1 𝑥 𝜑
2 iota5f.2 𝑥 𝐴
3 iota5f.3 ( ( 𝜑𝐴𝑉 ) → ( 𝜓𝑥 = 𝐴 ) )
4 2 nfel1 𝑥 𝐴𝑉
5 1 4 nfan 𝑥 ( 𝜑𝐴𝑉 )
6 5 3 alrimi ( ( 𝜑𝐴𝑉 ) → ∀ 𝑥 ( 𝜓𝑥 = 𝐴 ) )
7 2 nfeq2 𝑥 𝑦 = 𝐴
8 eqeq2 ( 𝑦 = 𝐴 → ( 𝑥 = 𝑦𝑥 = 𝐴 ) )
9 8 bibi2d ( 𝑦 = 𝐴 → ( ( 𝜓𝑥 = 𝑦 ) ↔ ( 𝜓𝑥 = 𝐴 ) ) )
10 7 9 albid ( 𝑦 = 𝐴 → ( ∀ 𝑥 ( 𝜓𝑥 = 𝑦 ) ↔ ∀ 𝑥 ( 𝜓𝑥 = 𝐴 ) ) )
11 eqeq2 ( 𝑦 = 𝐴 → ( ( ℩ 𝑥 𝜓 ) = 𝑦 ↔ ( ℩ 𝑥 𝜓 ) = 𝐴 ) )
12 10 11 imbi12d ( 𝑦 = 𝐴 → ( ( ∀ 𝑥 ( 𝜓𝑥 = 𝑦 ) → ( ℩ 𝑥 𝜓 ) = 𝑦 ) ↔ ( ∀ 𝑥 ( 𝜓𝑥 = 𝐴 ) → ( ℩ 𝑥 𝜓 ) = 𝐴 ) ) )
13 iotaval ( ∀ 𝑥 ( 𝜓𝑥 = 𝑦 ) → ( ℩ 𝑥 𝜓 ) = 𝑦 )
14 12 13 vtoclg ( 𝐴𝑉 → ( ∀ 𝑥 ( 𝜓𝑥 = 𝐴 ) → ( ℩ 𝑥 𝜓 ) = 𝐴 ) )
15 14 adantl ( ( 𝜑𝐴𝑉 ) → ( ∀ 𝑥 ( 𝜓𝑥 = 𝐴 ) → ( ℩ 𝑥 𝜓 ) = 𝐴 ) )
16 6 15 mpd ( ( 𝜑𝐴𝑉 ) → ( ℩ 𝑥 𝜓 ) = 𝐴 )