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Theorem bnj1534 31049
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1534.1 𝐷 = {𝑥𝐴 ∣ (𝐹𝑥) ≠ (𝐻𝑥)}
bnj1534.2 (𝑤𝐹 → ∀𝑥 𝑤𝐹)
Assertion
Ref Expression
bnj1534 𝐷 = {𝑧𝐴 ∣ (𝐹𝑧) ≠ (𝐻𝑧)}
Distinct variable groups:   𝑤,𝐴,𝑥,𝑧   𝑤,𝐹,𝑧   𝑤,𝐻,𝑥,𝑧
Allowed substitution hints:   𝐷(𝑥,𝑧,𝑤)   𝐹(𝑥)

Proof of Theorem bnj1534
StepHypRef Expression
1 bnj1534.1 . 2 𝐷 = {𝑥𝐴 ∣ (𝐹𝑥) ≠ (𝐻𝑥)}
2 nfcv 2793 . . 3 𝑥𝐴
3 nfcv 2793 . . 3 𝑧𝐴
4 nfv 1883 . . 3 𝑧(𝐹𝑥) ≠ (𝐻𝑥)
5 bnj1534.2 . . . . . 6 (𝑤𝐹 → ∀𝑥 𝑤𝐹)
65nfcii 2784 . . . . 5 𝑥𝐹
7 nfcv 2793 . . . . 5 𝑥𝑧
86, 7nffv 6236 . . . 4 𝑥(𝐹𝑧)
9 nfcv 2793 . . . 4 𝑥(𝐻𝑧)
108, 9nfne 2923 . . 3 𝑥(𝐹𝑧) ≠ (𝐻𝑧)
11 fveq2 6229 . . . 4 (𝑥 = 𝑧 → (𝐹𝑥) = (𝐹𝑧))
12 fveq2 6229 . . . 4 (𝑥 = 𝑧 → (𝐻𝑥) = (𝐻𝑧))
1311, 12neeq12d 2884 . . 3 (𝑥 = 𝑧 → ((𝐹𝑥) ≠ (𝐻𝑥) ↔ (𝐹𝑧) ≠ (𝐻𝑧)))
142, 3, 4, 10, 13cbvrab 3229 . 2 {𝑥𝐴 ∣ (𝐹𝑥) ≠ (𝐻𝑥)} = {𝑧𝐴 ∣ (𝐹𝑧) ≠ (𝐻𝑧)}
151, 14eqtri 2673 1 𝐷 = {𝑧𝐴 ∣ (𝐹𝑧) ≠ (𝐻𝑧)}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1521   = wceq 1523  wcel 2030  wne 2823  {crab 2945  cfv 5926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-iota 5889  df-fv 5934
This theorem is referenced by:  bnj1523  31265
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