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Theorem bnj1542 34702
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 7044 . . . . . 6 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ ∀𝑦𝐴 (𝐹𝑦) = (𝐺𝑦)))
54necon3abid 2967 . . . . 5 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐹𝐺 ↔ ¬ ∀𝑦𝐴 (𝐹𝑦) = (𝐺𝑦)))
6 df-ne 2931 . . . . . . 7 ((𝐹𝑦) ≠ (𝐺𝑦) ↔ ¬ (𝐹𝑦) = (𝐺𝑦))
76rexbii 3084 . . . . . 6 (∃𝑦𝐴 (𝐹𝑦) ≠ (𝐺𝑦) ↔ ∃𝑦𝐴 ¬ (𝐹𝑦) = (𝐺𝑦))
8 rexnal 3090 . . . . . 6 (∃𝑦𝐴 ¬ (𝐹𝑦) = (𝐺𝑦) ↔ ¬ ∀𝑦𝐴 (𝐹𝑦) = (𝐺𝑦))
97, 8bitri 274 . . . . 5 (∃𝑦𝐴 (𝐹𝑦) ≠ (𝐺𝑦) ↔ ¬ ∀𝑦𝐴 (𝐹𝑦) = (𝐺𝑦))
105, 9bitr4di 288 . . . 4 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐹𝐺 ↔ ∃𝑦𝐴 (𝐹𝑦) ≠ (𝐺𝑦)))
112, 3, 10syl2anc 582 . . 3 (𝜑 → (𝐹𝐺 ↔ ∃𝑦𝐴 (𝐹𝑦) ≠ (𝐺𝑦)))
121, 11mpbid 231 . 2 (𝜑 → ∃𝑦𝐴 (𝐹𝑦) ≠ (𝐺𝑦))
13 nfv 1910 . . 3 𝑦(𝐹𝑥) ≠ (𝐺𝑥)
14 bnj1542.4 . . . . . 6 (𝑤𝐹 → ∀𝑥 𝑤𝐹)
1514nfcii 2880 . . . . 5 𝑥𝐹
16 nfcv 2892 . . . . 5 𝑥𝑦
1715, 16nffv 6911 . . . 4 𝑥(𝐹𝑦)
18 nfcv 2892 . . . 4 𝑥(𝐺𝑦)
1917, 18nfne 3033 . . 3 𝑥(𝐹𝑦) ≠ (𝐺𝑦)
20 fveq2 6901 . . . 4 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
21 fveq2 6901 . . . 4 (𝑥 = 𝑦 → (𝐺𝑥) = (𝐺𝑦))
2220, 21neeq12d 2992 . . 3 (𝑥 = 𝑦 → ((𝐹𝑥) ≠ (𝐺𝑥) ↔ (𝐹𝑦) ≠ (𝐺𝑦)))
2313, 19, 22cbvrexw 3295 . 2 (∃𝑥𝐴 (𝐹𝑥) ≠ (𝐺𝑥) ↔ ∃𝑦𝐴 (𝐹𝑦) ≠ (𝐺𝑦))
2412, 23sylibr 233 1 (𝜑 → ∃𝑥𝐴 (𝐹𝑥) ≠ (𝐺𝑥))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394  wal 1532   = wceq 1534  wcel 2099  wne 2930  wral 3051  wrex 3060   Fn wfn 6549  cfv 6554
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6506  df-fun 6556  df-fn 6557  df-fv 6562
This theorem is referenced by:  bnj1523  34916
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