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Theorem tz6.12f 5123
 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 3520 . . . . 5 (z = y → ⟨A, z⟩ = ⟨A, y⟩)
21eleq1d 2084 . . . 4 (z = y → (⟨A, z 𝐹 ↔ ⟨A, y 𝐹))
3 tz6.12f.1 . . . . . . 7 y𝐹
43nfel2 2168 . . . . . 6 yA, z 𝐹
5 nfv 1398 . . . . . 6 zA, y 𝐹
64, 5, 2cbveu 1902 . . . . 5 (∃!zA, z 𝐹∃!yA, y 𝐹)
76a1i 9 . . . 4 (z = y → (∃!zA, z 𝐹∃!yA, y 𝐹))
82, 7anbi12d 445 . . 3 (z = y → ((⟨A, z 𝐹 ∃!zA, z 𝐹) ↔ (⟨A, y 𝐹 ∃!yA, y 𝐹)))
9 eqeq2 2027 . . 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 5122 . 2 ((⟨A, z 𝐹 ∃!zA, z 𝐹) → (𝐹A) = z)
1210, 11chvarv 1790 1 ((⟨A, y 𝐹 ∃!yA, y 𝐹) → (𝐹A) = y)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1226   ∈ wcel 1370  ∃!weu 1878  Ⅎwnfc 2143  ⟨cop 3349  ‘cfv 4825 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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000 This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-rex 2286  df-v 2533  df-sbc 2738  df-un 2895  df-sn 3352  df-pr 3353  df-op 3355  df-uni 3551  df-br 3735  df-iota 4790  df-fv 4833 This theorem is referenced by: (None)
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