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Theorem bnj1542 32028
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1542.1 (𝜑𝐹 Fn 𝐴)
bnj1542.2 (𝜑𝐺 Fn 𝐴)
bnj1542.3 (𝜑𝐹𝐺)
bnj1542.4 (𝑤𝐹 → ∀𝑥 𝑤𝐹)
Assertion
Ref Expression
bnj1542 (𝜑 → ∃𝑥𝐴 (𝐹𝑥) ≠ (𝐺𝑥))
Distinct variable groups:   𝑥,𝐴   𝑤,𝐹   𝑤,𝐺,𝑥
Allowed substitution hints:   𝜑(𝑥,𝑤)   𝐴(𝑤)   𝐹(𝑥)

Proof of Theorem bnj1542
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 bnj1542.3 . . 3 (𝜑𝐹𝐺)
2 bnj1542.1 . . . 4 (𝜑𝐹 Fn 𝐴)
3 bnj1542.2 . . . 4 (𝜑𝐺 Fn 𝐴)
4 eqfnfv 6794 . . . . . 6 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ ∀𝑦𝐴 (𝐹𝑦) = (𝐺𝑦)))
54necon3abid 3049 . . . . 5 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐹𝐺 ↔ ¬ ∀𝑦𝐴 (𝐹𝑦) = (𝐺𝑦)))
6 df-ne 3014 . . . . . . 7 ((𝐹𝑦) ≠ (𝐺𝑦) ↔ ¬ (𝐹𝑦) = (𝐺𝑦))
76rexbii 3244 . . . . . 6 (∃𝑦𝐴 (𝐹𝑦) ≠ (𝐺𝑦) ↔ ∃𝑦𝐴 ¬ (𝐹𝑦) = (𝐺𝑦))
8 rexnal 3235 . . . . . 6 (∃𝑦𝐴 ¬ (𝐹𝑦) = (𝐺𝑦) ↔ ¬ ∀𝑦𝐴 (𝐹𝑦) = (𝐺𝑦))
97, 8bitri 276 . . . . 5 (∃𝑦𝐴 (𝐹𝑦) ≠ (𝐺𝑦) ↔ ¬ ∀𝑦𝐴 (𝐹𝑦) = (𝐺𝑦))
105, 9syl6bbr 290 . . . 4 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐹𝐺 ↔ ∃𝑦𝐴 (𝐹𝑦) ≠ (𝐺𝑦)))
112, 3, 10syl2anc 584 . . 3 (𝜑 → (𝐹𝐺 ↔ ∃𝑦𝐴 (𝐹𝑦) ≠ (𝐺𝑦)))
121, 11mpbid 233 . 2 (𝜑 → ∃𝑦𝐴 (𝐹𝑦) ≠ (𝐺𝑦))
13 nfv 1906 . . 3 𝑦(𝐹𝑥) ≠ (𝐺𝑥)
14 bnj1542.4 . . . . . 6 (𝑤𝐹 → ∀𝑥 𝑤𝐹)
1514nfcii 2962 . . . . 5 𝑥𝐹
16 nfcv 2974 . . . . 5 𝑥𝑦
1715, 16nffv 6673 . . . 4 𝑥(𝐹𝑦)
18 nfcv 2974 . . . 4 𝑥(𝐺𝑦)
1917, 18nfne 3116 . . 3 𝑥(𝐹𝑦) ≠ (𝐺𝑦)
20 fveq2 6663 . . . 4 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
21 fveq2 6663 . . . 4 (𝑥 = 𝑦 → (𝐺𝑥) = (𝐺𝑦))
2220, 21neeq12d 3074 . . 3 (𝑥 = 𝑦 → ((𝐹𝑥) ≠ (𝐺𝑥) ↔ (𝐹𝑦) ≠ (𝐺𝑦)))
2313, 19, 22cbvrexw 3440 . 2 (∃𝑥𝐴 (𝐹𝑥) ≠ (𝐺𝑥) ↔ ∃𝑦𝐴 (𝐹𝑦) ≠ (𝐺𝑦))
2412, 23sylibr 235 1 (𝜑 → ∃𝑥𝐴 (𝐹𝑥) ≠ (𝐺𝑥))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wal 1526   = wceq 1528  wcel 2105  wne 3013  wral 3135  wrex 3136   Fn wfn 6343  cfv 6348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-fv 6356
This theorem is referenced by:  bnj1523  32240
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