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Theorem nfopd 3557
Description: Deduction version of bound-variable hypothesis builder nfop 3556. This shows how the deduction version of a not-free theorem such as nfop 3556 can be created from the corresponding not-free inference theorem. (Contributed by NM, 4-Feb-2008.)
Hypotheses
Ref Expression
nfopd.2 (φxA)
nfopd.3 (φxB)
Assertion
Ref Expression
nfopd (φxA, B⟩)

Proof of Theorem nfopd
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 nfaba1 2180 . . 3 x{zx z A}
2 nfaba1 2180 . . 3 x{zx z B}
31, 2nfop 3556 . 2 x⟨{zx z A}, {zx z B}⟩
4 nfopd.2 . . 3 (φxA)
5 nfopd.3 . . 3 (φxB)
6 nfnfc1 2178 . . . . 5 xxA
7 nfnfc1 2178 . . . . 5 xxB
86, 7nfan 1454 . . . 4 x(xA xB)
9 abidnf 2703 . . . . . 6 (xA → {zx z A} = A)
109adantr 261 . . . . 5 ((xA xB) → {zx z A} = A)
11 abidnf 2703 . . . . . 6 (xB → {zx z B} = B)
1211adantl 262 . . . . 5 ((xA xB) → {zx z B} = B)
1310, 12opeq12d 3548 . . . 4 ((xA xB) → ⟨{zx z A}, {zx z B}⟩ = ⟨A, B⟩)
148, 13nfceqdf 2174 . . 3 ((xA xB) → (x⟨{zx z A}, {zx z B}⟩ ↔ xA, B⟩))
154, 5, 14syl2anc 391 . 2 (φ → (x⟨{zx z A}, {zx z B}⟩ ↔ xA, B⟩))
163, 15mpbii 136 1 (φxA, B⟩)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wal 1240   = wceq 1242   wcel 1390  {cab 2023  wnfc 2162  cop 3370
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-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-sn 3373  df-pr 3374  df-op 3376
This theorem is referenced by:  nfbrd  3798  nfovd  5477
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