ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  tz6.12f Structured version   GIF version

Theorem tz6.12f 5145
Description: Function value, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 30-Aug-1999.)
Hypothesis
Ref Expression
tz6.12f.1 y𝐹
Assertion
Ref Expression
tz6.12f ((⟨A, y 𝐹 ∃!yA, y 𝐹) → (𝐹A) = y)
Distinct variable group:   y,A
Allowed substitution hint:   𝐹(y)

Proof of Theorem tz6.12f
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 opeq2 3541 . . . . 5 (z = y → ⟨A, z⟩ = ⟨A, y⟩)
21eleq1d 2103 . . . 4 (z = y → (⟨A, z 𝐹 ↔ ⟨A, y 𝐹))
3 tz6.12f.1 . . . . . . 7 y𝐹
43nfel2 2187 . . . . . 6 yA, z 𝐹
5 nfv 1418 . . . . . 6 zA, y 𝐹
64, 5, 2cbveu 1921 . . . . 5 (∃!zA, z 𝐹∃!yA, y 𝐹)
76a1i 9 . . . 4 (z = y → (∃!zA, z 𝐹∃!yA, y 𝐹))
82, 7anbi12d 442 . . 3 (z = y → ((⟨A, z 𝐹 ∃!zA, z 𝐹) ↔ (⟨A, y 𝐹 ∃!yA, y 𝐹)))
9 eqeq2 2046 . . 3 (z = y → ((𝐹A) = z ↔ (𝐹A) = y))
108, 9imbi12d 223 . 2 (z = y → (((⟨A, z 𝐹 ∃!zA, z 𝐹) → (𝐹A) = z) ↔ ((⟨A, y 𝐹 ∃!yA, y 𝐹) → (𝐹A) = y)))
11 tz6.12 5144 . 2 ((⟨A, z 𝐹 ∃!zA, z 𝐹) → (𝐹A) = z)
1210, 11chvarv 1809 1 ((⟨A, y 𝐹 ∃!yA, y 𝐹) → (𝐹A) = y)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242   wcel 1390  ∃!weu 1897  wnfc 2162  cop 3370  cfv 4845
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-iota 4810  df-fv 4853
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator