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Theorem nffvd 5282
Description: Deduction version of bound-variable hypothesis builder nffv 5280. (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 2230 . . 3 𝑥{𝑧 ∣ ∀𝑥 𝑧𝐹}
2 nfaba1 2230 . . 3 𝑥{𝑧 ∣ ∀𝑥 𝑧𝐴}
31, 2nffv 5280 . 2 𝑥({𝑧 ∣ ∀𝑥 𝑧𝐹}‘{𝑧 ∣ ∀𝑥 𝑧𝐴})
4 nffvd.2 . . 3 (𝜑𝑥𝐹)
5 nffvd.3 . . 3 (𝜑𝑥𝐴)
6 nfnfc1 2228 . . . . 5 𝑥𝑥𝐹
7 nfnfc1 2228 . . . . 5 𝑥𝑥𝐴
86, 7nfan 1500 . . . 4 𝑥(𝑥𝐹𝑥𝐴)
9 abidnf 2774 . . . . . 6 (𝑥𝐹 → {𝑧 ∣ ∀𝑥 𝑧𝐹} = 𝐹)
109adantr 270 . . . . 5 ((𝑥𝐹𝑥𝐴) → {𝑧 ∣ ∀𝑥 𝑧𝐹} = 𝐹)
11 abidnf 2774 . . . . . 6 (𝑥𝐴 → {𝑧 ∣ ∀𝑥 𝑧𝐴} = 𝐴)
1211adantl 271 . . . . 5 ((𝑥𝐹𝑥𝐴) → {𝑧 ∣ ∀𝑥 𝑧𝐴} = 𝐴)
1310, 12fveq12d 5277 . . . 4 ((𝑥𝐹𝑥𝐴) → ({𝑧 ∣ ∀𝑥 𝑧𝐹}‘{𝑧 ∣ ∀𝑥 𝑧𝐴}) = (𝐹𝐴))
148, 13nfceqdf 2224 . . 3 ((𝑥𝐹𝑥𝐴) → (𝑥({𝑧 ∣ ∀𝑥 𝑧𝐹}‘{𝑧 ∣ ∀𝑥 𝑧𝐴}) ↔ 𝑥(𝐹𝐴)))
154, 5, 14syl2anc 403 . 2 (𝜑 → (𝑥({𝑧 ∣ ∀𝑥 𝑧𝐹}‘{𝑧 ∣ ∀𝑥 𝑧𝐴}) ↔ 𝑥(𝐹𝐴)))
163, 15mpbii 146 1 (𝜑𝑥(𝐹𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wal 1285   = wceq 1287  wcel 1436  {cab 2071  wnfc 2212  cfv 4983
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-rex 2361  df-v 2617  df-un 2992  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3639  df-br 3823  df-iota 4948  df-fv 4991
This theorem is referenced by:  nfovd  5637
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