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Theorem eujustALT 2472
Description: Alternate proof of eujust 2471 illustrating the use of dvelim 2336. (Contributed by NM, 11-Mar-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
eujustALT (∃𝑦𝑥(𝜑𝑥 = 𝑦) ↔ ∃𝑧𝑥(𝜑𝑥 = 𝑧))
Distinct variable groups:   𝑥,𝑦   𝑥,𝑧   𝜑,𝑦   𝜑,𝑧
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem eujustALT
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 equequ2 1950 . . . . . 6 (𝑦 = 𝑧 → (𝑥 = 𝑦𝑥 = 𝑧))
21bibi2d 332 . . . . 5 (𝑦 = 𝑧 → ((𝜑𝑥 = 𝑦) ↔ (𝜑𝑥 = 𝑧)))
32albidv 1846 . . . 4 (𝑦 = 𝑧 → (∀𝑥(𝜑𝑥 = 𝑦) ↔ ∀𝑥(𝜑𝑥 = 𝑧)))
43sps 2053 . . 3 (∀𝑦 𝑦 = 𝑧 → (∀𝑥(𝜑𝑥 = 𝑦) ↔ ∀𝑥(𝜑𝑥 = 𝑧)))
54drex1 2326 . 2 (∀𝑦 𝑦 = 𝑧 → (∃𝑦𝑥(𝜑𝑥 = 𝑦) ↔ ∃𝑧𝑥(𝜑𝑥 = 𝑧)))
6 hbnae 2316 . . . . . 6 (¬ ∀𝑦 𝑦 = 𝑧 → ∀𝑦 ¬ ∀𝑦 𝑦 = 𝑧)
7 hbnae 2316 . . . . . 6 (¬ ∀𝑦 𝑦 = 𝑧 → ∀𝑧 ¬ ∀𝑦 𝑦 = 𝑧)
86, 7alrimih 1748 . . . . 5 (¬ ∀𝑦 𝑦 = 𝑧 → ∀𝑦𝑧 ¬ ∀𝑦 𝑦 = 𝑧)
9 ax-5 1836 . . . . . . . 8 (¬ ∀𝑥(𝜑𝑥 = 𝑤) → ∀𝑧 ¬ ∀𝑥(𝜑𝑥 = 𝑤))
10 equequ2 1950 . . . . . . . . . . 11 (𝑤 = 𝑦 → (𝑥 = 𝑤𝑥 = 𝑦))
1110bibi2d 332 . . . . . . . . . 10 (𝑤 = 𝑦 → ((𝜑𝑥 = 𝑤) ↔ (𝜑𝑥 = 𝑦)))
1211albidv 1846 . . . . . . . . 9 (𝑤 = 𝑦 → (∀𝑥(𝜑𝑥 = 𝑤) ↔ ∀𝑥(𝜑𝑥 = 𝑦)))
1312notbid 308 . . . . . . . 8 (𝑤 = 𝑦 → (¬ ∀𝑥(𝜑𝑥 = 𝑤) ↔ ¬ ∀𝑥(𝜑𝑥 = 𝑦)))
149, 13dvelim 2336 . . . . . . 7 (¬ ∀𝑧 𝑧 = 𝑦 → (¬ ∀𝑥(𝜑𝑥 = 𝑦) → ∀𝑧 ¬ ∀𝑥(𝜑𝑥 = 𝑦)))
1514naecoms 2312 . . . . . 6 (¬ ∀𝑦 𝑦 = 𝑧 → (¬ ∀𝑥(𝜑𝑥 = 𝑦) → ∀𝑧 ¬ ∀𝑥(𝜑𝑥 = 𝑦)))
16 ax-5 1836 . . . . . . 7 (¬ ∀𝑥(𝜑𝑥 = 𝑤) → ∀𝑦 ¬ ∀𝑥(𝜑𝑥 = 𝑤))
17 equequ2 1950 . . . . . . . . . 10 (𝑤 = 𝑧 → (𝑥 = 𝑤𝑥 = 𝑧))
1817bibi2d 332 . . . . . . . . 9 (𝑤 = 𝑧 → ((𝜑𝑥 = 𝑤) ↔ (𝜑𝑥 = 𝑧)))
1918albidv 1846 . . . . . . . 8 (𝑤 = 𝑧 → (∀𝑥(𝜑𝑥 = 𝑤) ↔ ∀𝑥(𝜑𝑥 = 𝑧)))
2019notbid 308 . . . . . . 7 (𝑤 = 𝑧 → (¬ ∀𝑥(𝜑𝑥 = 𝑤) ↔ ¬ ∀𝑥(𝜑𝑥 = 𝑧)))
2116, 20dvelim 2336 . . . . . 6 (¬ ∀𝑦 𝑦 = 𝑧 → (¬ ∀𝑥(𝜑𝑥 = 𝑧) → ∀𝑦 ¬ ∀𝑥(𝜑𝑥 = 𝑧)))
223notbid 308 . . . . . . 7 (𝑦 = 𝑧 → (¬ ∀𝑥(𝜑𝑥 = 𝑦) ↔ ¬ ∀𝑥(𝜑𝑥 = 𝑧)))
2322a1i 11 . . . . . 6 (¬ ∀𝑦 𝑦 = 𝑧 → (𝑦 = 𝑧 → (¬ ∀𝑥(𝜑𝑥 = 𝑦) ↔ ¬ ∀𝑥(𝜑𝑥 = 𝑧))))
2415, 21, 23cbv2h 2268 . . . . 5 (∀𝑦𝑧 ¬ ∀𝑦 𝑦 = 𝑧 → (∀𝑦 ¬ ∀𝑥(𝜑𝑥 = 𝑦) ↔ ∀𝑧 ¬ ∀𝑥(𝜑𝑥 = 𝑧)))
258, 24syl 17 . . . 4 (¬ ∀𝑦 𝑦 = 𝑧 → (∀𝑦 ¬ ∀𝑥(𝜑𝑥 = 𝑦) ↔ ∀𝑧 ¬ ∀𝑥(𝜑𝑥 = 𝑧)))
2625notbid 308 . . 3 (¬ ∀𝑦 𝑦 = 𝑧 → (¬ ∀𝑦 ¬ ∀𝑥(𝜑𝑥 = 𝑦) ↔ ¬ ∀𝑧 ¬ ∀𝑥(𝜑𝑥 = 𝑧)))
27 df-ex 1702 . . 3 (∃𝑦𝑥(𝜑𝑥 = 𝑦) ↔ ¬ ∀𝑦 ¬ ∀𝑥(𝜑𝑥 = 𝑦))
28 df-ex 1702 . . 3 (∃𝑧𝑥(𝜑𝑥 = 𝑧) ↔ ¬ ∀𝑧 ¬ ∀𝑥(𝜑𝑥 = 𝑧))
2926, 27, 283bitr4g 303 . 2 (¬ ∀𝑦 𝑦 = 𝑧 → (∃𝑦𝑥(𝜑𝑥 = 𝑦) ↔ ∃𝑧𝑥(𝜑𝑥 = 𝑧)))
305, 29pm2.61i 176 1 (∃𝑦𝑥(𝜑𝑥 = 𝑦) ↔ ∃𝑧𝑥(𝜑𝑥 = 𝑧))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wal 1478  wex 1701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707
This theorem is referenced by: (None)
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