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Theorem nffvd 6112
Description: Deduction version of bound-variable hypothesis builder nffv 6110. (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 2756 . . 3 𝑥{𝑧 ∣ ∀𝑥 𝑧𝐹}
2 nfaba1 2756 . . 3 𝑥{𝑧 ∣ ∀𝑥 𝑧𝐴}
31, 2nffv 6110 . 2 𝑥({𝑧 ∣ ∀𝑥 𝑧𝐹}‘{𝑧 ∣ ∀𝑥 𝑧𝐴})
4 nffvd.2 . . 3 (𝜑𝑥𝐹)
5 nffvd.3 . . 3 (𝜑𝑥𝐴)
6 nfnfc1 2754 . . . . 5 𝑥𝑥𝐹
7 nfnfc1 2754 . . . . 5 𝑥𝑥𝐴
86, 7nfan 1816 . . . 4 𝑥(𝑥𝐹𝑥𝐴)
9 abidnf 3342 . . . . . 6 (𝑥𝐹 → {𝑧 ∣ ∀𝑥 𝑧𝐹} = 𝐹)
109adantr 480 . . . . 5 ((𝑥𝐹𝑥𝐴) → {𝑧 ∣ ∀𝑥 𝑧𝐹} = 𝐹)
11 abidnf 3342 . . . . . 6 (𝑥𝐴 → {𝑧 ∣ ∀𝑥 𝑧𝐴} = 𝐴)
1211adantl 481 . . . . 5 ((𝑥𝐹𝑥𝐴) → {𝑧 ∣ ∀𝑥 𝑧𝐴} = 𝐴)
1310, 12fveq12d 6109 . . . 4 ((𝑥𝐹𝑥𝐴) → ({𝑧 ∣ ∀𝑥 𝑧𝐹}‘{𝑧 ∣ ∀𝑥 𝑧𝐴}) = (𝐹𝐴))
148, 13nfceqdf 2747 . . 3 ((𝑥𝐹𝑥𝐴) → (𝑥({𝑧 ∣ ∀𝑥 𝑧𝐹}‘{𝑧 ∣ ∀𝑥 𝑧𝐴}) ↔ 𝑥(𝐹𝐴)))
154, 5, 14syl2anc 691 . 2 (𝜑 → (𝑥({𝑧 ∣ ∀𝑥 𝑧𝐹}‘{𝑧 ∣ ∀𝑥 𝑧𝐴}) ↔ 𝑥(𝐹𝐴)))
163, 15mpbii 222 1 (𝜑𝑥(𝐹𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383  wal 1473   = wceq 1475  wcel 1977  {cab 2596  wnfc 2738  cfv 5804
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812
This theorem is referenced by:  nfovd  6574  nfixp  7813
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