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Theorem bnj1534 30623
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 2767 . . 3 𝑥𝐴
3 nfcv 2767 . . 3 𝑧𝐴
4 nfv 1845 . . 3 𝑧(𝐹𝑥) ≠ (𝐻𝑥)
5 bnj1534.2 . . . . . 6 (𝑤𝐹 → ∀𝑥 𝑤𝐹)
65nfcii 2758 . . . . 5 𝑥𝐹
7 nfcv 2767 . . . . 5 𝑥𝑧
86, 7nffv 6157 . . . 4 𝑥(𝐹𝑧)
9 nfcv 2767 . . . 4 𝑥(𝐻𝑧)
108, 9nfne 2896 . . 3 𝑥(𝐹𝑧) ≠ (𝐻𝑧)
11 fveq2 6150 . . . 4 (𝑥 = 𝑧 → (𝐹𝑥) = (𝐹𝑧))
12 fveq2 6150 . . . 4 (𝑥 = 𝑧 → (𝐻𝑥) = (𝐻𝑧))
1311, 12neeq12d 2857 . . 3 (𝑥 = 𝑧 → ((𝐹𝑥) ≠ (𝐻𝑥) ↔ (𝐹𝑧) ≠ (𝐻𝑧)))
142, 3, 4, 10, 13cbvrab 3189 . 2 {𝑥𝐴 ∣ (𝐹𝑥) ≠ (𝐻𝑥)} = {𝑧𝐴 ∣ (𝐹𝑧) ≠ (𝐻𝑧)}
151, 14eqtri 2648 1 𝐷 = {𝑧𝐴 ∣ (𝐹𝑧) ≠ (𝐻𝑧)}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1478   = wceq 1480  wcel 1992  wne 2796  {crab 2916  cfv 5850
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-iota 5813  df-fv 5858
This theorem is referenced by:  bnj1523  30839
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