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Theorem bnj1534 32199
 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 2979 . . 3 𝑥𝐴
3 nfcv 2979 . . 3 𝑧𝐴
4 nfv 1915 . . 3 𝑧(𝐹𝑥) ≠ (𝐻𝑥)
5 bnj1534.2 . . . . . 6 (𝑤𝐹 → ∀𝑥 𝑤𝐹)
65nfcii 2964 . . . . 5 𝑥𝐹
7 nfcv 2979 . . . . 5 𝑥𝑧
86, 7nffv 6662 . . . 4 𝑥(𝐹𝑧)
9 nfcv 2979 . . . 4 𝑥(𝐻𝑧)
108, 9nfne 3111 . . 3 𝑥(𝐹𝑧) ≠ (𝐻𝑧)
11 fveq2 6652 . . . 4 (𝑥 = 𝑧 → (𝐹𝑥) = (𝐹𝑧))
12 fveq2 6652 . . . 4 (𝑥 = 𝑧 → (𝐻𝑥) = (𝐻𝑧))
1311, 12neeq12d 3072 . . 3 (𝑥 = 𝑧 → ((𝐹𝑥) ≠ (𝐻𝑥) ↔ (𝐹𝑧) ≠ (𝐻𝑧)))
142, 3, 4, 10, 13cbvrabw 3465 . 2 {𝑥𝐴 ∣ (𝐹𝑥) ≠ (𝐻𝑥)} = {𝑧𝐴 ∣ (𝐹𝑧) ≠ (𝐻𝑧)}
151, 14eqtri 2845 1 𝐷 = {𝑧𝐴 ∣ (𝐹𝑧) ≠ (𝐻𝑧)}
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1536   = wceq 1538   ∈ wcel 2114   ≠ wne 3011  {crab 3134  ‘cfv 6334 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-iota 6293  df-fv 6342 This theorem is referenced by:  bnj1523  32417
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