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Theorem bnj1542 32845
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 6901 . . . . . 6 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ ∀𝑦𝐴 (𝐹𝑦) = (𝐺𝑦)))
54necon3abid 2980 . . . . 5 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐹𝐺 ↔ ¬ ∀𝑦𝐴 (𝐹𝑦) = (𝐺𝑦)))
6 df-ne 2944 . . . . . . 7 ((𝐹𝑦) ≠ (𝐺𝑦) ↔ ¬ (𝐹𝑦) = (𝐺𝑦))
76rexbii 3179 . . . . . 6 (∃𝑦𝐴 (𝐹𝑦) ≠ (𝐺𝑦) ↔ ∃𝑦𝐴 ¬ (𝐹𝑦) = (𝐺𝑦))
8 rexnal 3167 . . . . . 6 (∃𝑦𝐴 ¬ (𝐹𝑦) = (𝐺𝑦) ↔ ¬ ∀𝑦𝐴 (𝐹𝑦) = (𝐺𝑦))
97, 8bitri 274 . . . . 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 231 . 2 (𝜑 → ∃𝑦𝐴 (𝐹𝑦) ≠ (𝐺𝑦))
13 nfv 1917 . . 3 𝑦(𝐹𝑥) ≠ (𝐺𝑥)
14 bnj1542.4 . . . . . 6 (𝑤𝐹 → ∀𝑥 𝑤𝐹)
1514nfcii 2891 . . . . 5 𝑥𝐹
16 nfcv 2907 . . . . 5 𝑥𝑦
1715, 16nffv 6776 . . . 4 𝑥(𝐹𝑦)
18 nfcv 2907 . . . 4 𝑥(𝐺𝑦)
1917, 18nfne 3045 . . 3 𝑥(𝐹𝑦) ≠ (𝐺𝑦)
20 fveq2 6766 . . . 4 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
21 fveq2 6766 . . . 4 (𝑥 = 𝑦 → (𝐺𝑥) = (𝐺𝑦))
2220, 21neeq12d 3005 . . 3 (𝑥 = 𝑦 → ((𝐹𝑥) ≠ (𝐺𝑥) ↔ (𝐹𝑦) ≠ (𝐺𝑦)))
2313, 19, 22cbvrexw 3372 . 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 396  wal 1537   = wceq 1539  wcel 2106  wne 2943  wral 3064  wrex 3065   Fn wfn 6421  cfv 6426
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5221  ax-nul 5228  ax-pr 5350
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3431  df-sbc 3716  df-csb 3832  df-dif 3889  df-un 3891  df-in 3893  df-ss 3903  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5074  df-opab 5136  df-mpt 5157  df-id 5484  df-xp 5590  df-rel 5591  df-cnv 5592  df-co 5593  df-dm 5594  df-rn 5595  df-res 5596  df-ima 5597  df-iota 6384  df-fun 6428  df-fn 6429  df-fv 6434
This theorem is referenced by:  bnj1523  33059
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