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Theorem nffvd 5128
Description: Deduction version of bound-variable hypothesis builder nffv 5126. (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 15-Oct-2016.)
Hypotheses
Ref Expression
nffvd.2 (φx𝐹)
nffvd.3 (φxA)
Assertion
Ref Expression
nffvd (φx(𝐹A))

Proof of Theorem nffvd
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 nfaba1 2180 . . 3 x{zx z 𝐹}
2 nfaba1 2180 . . 3 x{zx z A}
31, 2nffv 5126 . 2 x({zx z 𝐹}‘{zx z A})
4 nffvd.2 . . 3 (φx𝐹)
5 nffvd.3 . . 3 (φxA)
6 nfnfc1 2178 . . . . 5 xx𝐹
7 nfnfc1 2178 . . . . 5 xxA
86, 7nfan 1454 . . . 4 x(x𝐹 xA)
9 abidnf 2703 . . . . . 6 (x𝐹 → {zx z 𝐹} = 𝐹)
109adantr 261 . . . . 5 ((x𝐹 xA) → {zx z 𝐹} = 𝐹)
11 abidnf 2703 . . . . . 6 (xA → {zx z A} = A)
1211adantl 262 . . . . 5 ((x𝐹 xA) → {zx z A} = A)
1310, 12fveq12d 5125 . . . 4 ((x𝐹 xA) → ({zx z 𝐹}‘{zx z A}) = (𝐹A))
148, 13nfceqdf 2174 . . 3 ((x𝐹 xA) → (x({zx z 𝐹}‘{zx z A}) ↔ x(𝐹A)))
154, 5, 14syl2anc 391 . 2 (φ → (x({zx z 𝐹}‘{zx z A}) ↔ x(𝐹A)))
163, 15mpbii 136 1 (φx(𝐹A))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wal 1240   = wceq 1242   wcel 1390  {cab 2023  wnfc 2162  cfv 4844
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-rex 2306  df-v 2553  df-un 2916  df-sn 3372  df-pr 3373  df-op 3375  df-uni 3571  df-br 3755  df-iota 4809  df-fv 4852
This theorem is referenced by:  nfovd  5474
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