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Theorem bnj1542 32122
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 6795 . . . . . 6 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ ∀𝑦𝐴 (𝐹𝑦) = (𝐺𝑦)))
54necon3abid 3050 . . . . 5 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐹𝐺 ↔ ¬ ∀𝑦𝐴 (𝐹𝑦) = (𝐺𝑦)))
6 df-ne 3015 . . . . . . 7 ((𝐹𝑦) ≠ (𝐺𝑦) ↔ ¬ (𝐹𝑦) = (𝐺𝑦))
76rexbii 3245 . . . . . 6 (∃𝑦𝐴 (𝐹𝑦) ≠ (𝐺𝑦) ↔ ∃𝑦𝐴 ¬ (𝐹𝑦) = (𝐺𝑦))
8 rexnal 3236 . . . . . 6 (∃𝑦𝐴 ¬ (𝐹𝑦) = (𝐺𝑦) ↔ ¬ ∀𝑦𝐴 (𝐹𝑦) = (𝐺𝑦))
97, 8bitri 277 . . . . 5 (∃𝑦𝐴 (𝐹𝑦) ≠ (𝐺𝑦) ↔ ¬ ∀𝑦𝐴 (𝐹𝑦) = (𝐺𝑦))
105, 9syl6bbr 291 . . . 4 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐹𝐺 ↔ ∃𝑦𝐴 (𝐹𝑦) ≠ (𝐺𝑦)))
112, 3, 10syl2anc 586 . . 3 (𝜑 → (𝐹𝐺 ↔ ∃𝑦𝐴 (𝐹𝑦) ≠ (𝐺𝑦)))
121, 11mpbid 234 . 2 (𝜑 → ∃𝑦𝐴 (𝐹𝑦) ≠ (𝐺𝑦))
13 nfv 1909 . . 3 𝑦(𝐹𝑥) ≠ (𝐺𝑥)
14 bnj1542.4 . . . . . 6 (𝑤𝐹 → ∀𝑥 𝑤𝐹)
1514nfcii 2963 . . . . 5 𝑥𝐹
16 nfcv 2975 . . . . 5 𝑥𝑦
1715, 16nffv 6673 . . . 4 𝑥(𝐹𝑦)
18 nfcv 2975 . . . 4 𝑥(𝐺𝑦)
1917, 18nfne 3117 . . 3 𝑥(𝐹𝑦) ≠ (𝐺𝑦)
20 fveq2 6663 . . . 4 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
21 fveq2 6663 . . . 4 (𝑥 = 𝑦 → (𝐺𝑥) = (𝐺𝑦))
2220, 21neeq12d 3075 . . 3 (𝑥 = 𝑦 → ((𝐹𝑥) ≠ (𝐺𝑥) ↔ (𝐹𝑦) ≠ (𝐺𝑦)))
2313, 19, 22cbvrexw 3441 . 2 (∃𝑥𝐴 (𝐹𝑥) ≠ (𝐺𝑥) ↔ ∃𝑦𝐴 (𝐹𝑦) ≠ (𝐺𝑦))
2412, 23sylibr 236 1 (𝜑 → ∃𝑥𝐴 (𝐹𝑥) ≠ (𝐺𝑥))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  wa 398  wal 1529   = wceq 1531  wcel 2108  wne 3014  wral 3136  wrex 3137   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 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  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  32336
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