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Theorem bnj1534 34615
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 2891 . . 3 𝑥𝐴
3 nfcv 2891 . . 3 𝑧𝐴
4 nfv 1909 . . 3 𝑧(𝐹𝑥) ≠ (𝐻𝑥)
5 bnj1534.2 . . . . . 6 (𝑤𝐹 → ∀𝑥 𝑤𝐹)
65nfcii 2879 . . . . 5 𝑥𝐹
7 nfcv 2891 . . . . 5 𝑥𝑧
86, 7nffv 6906 . . . 4 𝑥(𝐹𝑧)
9 nfcv 2891 . . . 4 𝑥(𝐻𝑧)
108, 9nfne 3032 . . 3 𝑥(𝐹𝑧) ≠ (𝐻𝑧)
11 fveq2 6896 . . . 4 (𝑥 = 𝑧 → (𝐹𝑥) = (𝐹𝑧))
12 fveq2 6896 . . . 4 (𝑥 = 𝑧 → (𝐻𝑥) = (𝐻𝑧))
1311, 12neeq12d 2991 . . 3 (𝑥 = 𝑧 → ((𝐹𝑥) ≠ (𝐻𝑥) ↔ (𝐹𝑧) ≠ (𝐻𝑧)))
142, 3, 4, 10, 13cbvrabw 3455 . 2 {𝑥𝐴 ∣ (𝐹𝑥) ≠ (𝐻𝑥)} = {𝑧𝐴 ∣ (𝐹𝑧) ≠ (𝐻𝑧)}
151, 14eqtri 2753 1 𝐷 = {𝑧𝐴 ∣ (𝐹𝑧) ≠ (𝐻𝑧)}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1531   = wceq 1533  wcel 2098  wne 2929  {crab 3418  cfv 6549
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-iota 6501  df-fv 6557
This theorem is referenced by:  bnj1523  34833
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