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Theorem bnj1534 31772
 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 2932 . . 3 𝑥𝐴
3 nfcv 2932 . . 3 𝑧𝐴
4 nfv 1873 . . 3 𝑧(𝐹𝑥) ≠ (𝐻𝑥)
5 bnj1534.2 . . . . . 6 (𝑤𝐹 → ∀𝑥 𝑤𝐹)
65nfcii 2920 . . . . 5 𝑥𝐹
7 nfcv 2932 . . . . 5 𝑥𝑧
86, 7nffv 6511 . . . 4 𝑥(𝐹𝑧)
9 nfcv 2932 . . . 4 𝑥(𝐻𝑧)
108, 9nfne 3070 . . 3 𝑥(𝐹𝑧) ≠ (𝐻𝑧)
11 fveq2 6501 . . . 4 (𝑥 = 𝑧 → (𝐹𝑥) = (𝐹𝑧))
12 fveq2 6501 . . . 4 (𝑥 = 𝑧 → (𝐻𝑥) = (𝐻𝑧))
1311, 12neeq12d 3028 . . 3 (𝑥 = 𝑧 → ((𝐹𝑥) ≠ (𝐻𝑥) ↔ (𝐹𝑧) ≠ (𝐻𝑧)))
142, 3, 4, 10, 13cbvrab 3411 . 2 {𝑥𝐴 ∣ (𝐹𝑥) ≠ (𝐻𝑥)} = {𝑧𝐴 ∣ (𝐹𝑧) ≠ (𝐻𝑧)}
151, 14eqtri 2802 1 𝐷 = {𝑧𝐴 ∣ (𝐹𝑧) ≠ (𝐻𝑧)}
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1505   = wceq 1507   ∈ wcel 2050   ≠ wne 2967  {crab 3092  ‘cfv 6190 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-rab 3097  df-v 3417  df-dif 3834  df-un 3836  df-in 3838  df-ss 3845  df-nul 4181  df-if 4352  df-sn 4443  df-pr 4445  df-op 4449  df-uni 4714  df-br 4931  df-iota 6154  df-fv 6198 This theorem is referenced by:  bnj1523  31988
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