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Theorem bnj1542 34871
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 7051 . . . . . 6 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ ∀𝑦𝐴 (𝐹𝑦) = (𝐺𝑦)))
54necon3abid 2977 . . . . 5 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐹𝐺 ↔ ¬ ∀𝑦𝐴 (𝐹𝑦) = (𝐺𝑦)))
6 df-ne 2941 . . . . . . 7 ((𝐹𝑦) ≠ (𝐺𝑦) ↔ ¬ (𝐹𝑦) = (𝐺𝑦))
76rexbii 3094 . . . . . 6 (∃𝑦𝐴 (𝐹𝑦) ≠ (𝐺𝑦) ↔ ∃𝑦𝐴 ¬ (𝐹𝑦) = (𝐺𝑦))
8 rexnal 3100 . . . . . 6 (∃𝑦𝐴 ¬ (𝐹𝑦) = (𝐺𝑦) ↔ ¬ ∀𝑦𝐴 (𝐹𝑦) = (𝐺𝑦))
97, 8bitri 275 . . . . 5 (∃𝑦𝐴 (𝐹𝑦) ≠ (𝐺𝑦) ↔ ¬ ∀𝑦𝐴 (𝐹𝑦) = (𝐺𝑦))
105, 9bitr4di 289 . . . 4 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐹𝐺 ↔ ∃𝑦𝐴 (𝐹𝑦) ≠ (𝐺𝑦)))
112, 3, 10syl2anc 584 . . 3 (𝜑 → (𝐹𝐺 ↔ ∃𝑦𝐴 (𝐹𝑦) ≠ (𝐺𝑦)))
121, 11mpbid 232 . 2 (𝜑 → ∃𝑦𝐴 (𝐹𝑦) ≠ (𝐺𝑦))
13 nfv 1914 . . 3 𝑦(𝐹𝑥) ≠ (𝐺𝑥)
14 bnj1542.4 . . . . . 6 (𝑤𝐹 → ∀𝑥 𝑤𝐹)
1514nfcii 2894 . . . . 5 𝑥𝐹
16 nfcv 2905 . . . . 5 𝑥𝑦
1715, 16nffv 6916 . . . 4 𝑥(𝐹𝑦)
18 nfcv 2905 . . . 4 𝑥(𝐺𝑦)
1917, 18nfne 3043 . . 3 𝑥(𝐹𝑦) ≠ (𝐺𝑦)
20 fveq2 6906 . . . 4 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
21 fveq2 6906 . . . 4 (𝑥 = 𝑦 → (𝐺𝑥) = (𝐺𝑦))
2220, 21neeq12d 3002 . . 3 (𝑥 = 𝑦 → ((𝐹𝑥) ≠ (𝐺𝑥) ↔ (𝐹𝑦) ≠ (𝐺𝑦)))
2313, 19, 22cbvrexw 3307 . 2 (∃𝑥𝐴 (𝐹𝑥) ≠ (𝐺𝑥) ↔ ∃𝑦𝐴 (𝐹𝑦) ≠ (𝐺𝑦))
2412, 23sylibr 234 1 (𝜑 → ∃𝑥𝐴 (𝐹𝑥) ≠ (𝐺𝑥))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wal 1538   = wceq 1540  wcel 2108  wne 2940  wral 3061  wrex 3070   Fn wfn 6556  cfv 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-fv 6569
This theorem is referenced by:  bnj1523  35085
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