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Theorem nffvd 6675
Description: Deduction version of bound-variable hypothesis builder nffv 6673. (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 2984 . . 3 𝑥{𝑧 ∣ ∀𝑥 𝑧𝐹}
2 nfaba1 2984 . . 3 𝑥{𝑧 ∣ ∀𝑥 𝑧𝐴}
31, 2nffv 6673 . 2 𝑥({𝑧 ∣ ∀𝑥 𝑧𝐹}‘{𝑧 ∣ ∀𝑥 𝑧𝐴})
4 nffvd.2 . . 3 (𝜑𝑥𝐹)
5 nffvd.3 . . 3 (𝜑𝑥𝐴)
6 nfnfc1 2978 . . . . 5 𝑥𝑥𝐹
7 nfnfc1 2978 . . . . 5 𝑥𝑥𝐴
86, 7nfan 1893 . . . 4 𝑥(𝑥𝐹𝑥𝐴)
9 abidnf 3692 . . . . . 6 (𝑥𝐹 → {𝑧 ∣ ∀𝑥 𝑧𝐹} = 𝐹)
109adantr 483 . . . . 5 ((𝑥𝐹𝑥𝐴) → {𝑧 ∣ ∀𝑥 𝑧𝐹} = 𝐹)
11 abidnf 3692 . . . . . 6 (𝑥𝐴 → {𝑧 ∣ ∀𝑥 𝑧𝐴} = 𝐴)
1211adantl 484 . . . . 5 ((𝑥𝐹𝑥𝐴) → {𝑧 ∣ ∀𝑥 𝑧𝐴} = 𝐴)
1310, 12fveq12d 6670 . . . 4 ((𝑥𝐹𝑥𝐴) → ({𝑧 ∣ ∀𝑥 𝑧𝐹}‘{𝑧 ∣ ∀𝑥 𝑧𝐴}) = (𝐹𝐴))
148, 13nfceqdf 2970 . . 3 ((𝑥𝐹𝑥𝐴) → (𝑥({𝑧 ∣ ∀𝑥 𝑧𝐹}‘{𝑧 ∣ ∀𝑥 𝑧𝐴}) ↔ 𝑥(𝐹𝐴)))
154, 5, 14syl2anc 586 . 2 (𝜑 → (𝑥({𝑧 ∣ ∀𝑥 𝑧𝐹}‘{𝑧 ∣ ∀𝑥 𝑧𝐴}) ↔ 𝑥(𝐹𝐴)))
163, 15mpbii 235 1 (𝜑𝑥(𝐹𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  wal 1528   = wceq 1530  wcel 2107  {cab 2797  wnfc 2959  cfv 6348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-iota 6307  df-fv 6356
This theorem is referenced by:  nfovd  7177  nfixpw  8472  nfixp  8473
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