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Theorem nffvd 6913
Description: Deduction version of bound-variable hypothesis builder nffv 6911. (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 2900 . . 3 𝑥{𝑧 ∣ ∀𝑥 𝑧𝐹}
2 nfaba1 2900 . . 3 𝑥{𝑧 ∣ ∀𝑥 𝑧𝐴}
31, 2nffv 6911 . 2 𝑥({𝑧 ∣ ∀𝑥 𝑧𝐹}‘{𝑧 ∣ ∀𝑥 𝑧𝐴})
4 nffvd.2 . . 3 (𝜑𝑥𝐹)
5 nffvd.3 . . 3 (𝜑𝑥𝐴)
6 nfnfc1 2895 . . . . 5 𝑥𝑥𝐹
7 nfnfc1 2895 . . . . 5 𝑥𝑥𝐴
86, 7nfan 1895 . . . 4 𝑥(𝑥𝐹𝑥𝐴)
9 abidnf 3696 . . . . . 6 (𝑥𝐹 → {𝑧 ∣ ∀𝑥 𝑧𝐹} = 𝐹)
109adantr 479 . . . . 5 ((𝑥𝐹𝑥𝐴) → {𝑧 ∣ ∀𝑥 𝑧𝐹} = 𝐹)
11 abidnf 3696 . . . . . 6 (𝑥𝐴 → {𝑧 ∣ ∀𝑥 𝑧𝐴} = 𝐴)
1211adantl 480 . . . . 5 ((𝑥𝐹𝑥𝐴) → {𝑧 ∣ ∀𝑥 𝑧𝐴} = 𝐴)
1310, 12fveq12d 6908 . . . 4 ((𝑥𝐹𝑥𝐴) → ({𝑧 ∣ ∀𝑥 𝑧𝐹}‘{𝑧 ∣ ∀𝑥 𝑧𝐴}) = (𝐹𝐴))
148, 13nfceqdf 2887 . . 3 ((𝑥𝐹𝑥𝐴) → (𝑥({𝑧 ∣ ∀𝑥 𝑧𝐹}‘{𝑧 ∣ ∀𝑥 𝑧𝐴}) ↔ 𝑥(𝐹𝐴)))
154, 5, 14syl2anc 582 . 2 (𝜑 → (𝑥({𝑧 ∣ ∀𝑥 𝑧𝐹}‘{𝑧 ∣ ∀𝑥 𝑧𝐴}) ↔ 𝑥(𝐹𝐴)))
163, 15mpbii 232 1 (𝜑𝑥(𝐹𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  wal 1532   = wceq 1534  wcel 2099  {cab 2703  wnfc 2876  cfv 6554
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-br 5154  df-iota 6506  df-fv 6562
This theorem is referenced by:  nfovd  7453  nfixpw  8945  nfixp  8946  bj-gabima  36646
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