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Theorem axnulALT 4820
Description: Alternate proof of axnul 4821, proved from propositional calculus, ax-gen 1762, ax-4 1777, sp 2091, and ax-rep 4804. To check this, replace sp 2091 with the obsolete axiom ax-c5 34487 in the proof of axnulALT 4820 and type the Metamath command 'SHOW TRACEBACK axnulALT / AXIOMS'. (Contributed by Jeff Hoffman, 3-Feb-2008.) (Proof shortened by Mario Carneiro, 17-Nov-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
axnulALT 𝑥𝑦 ¬ 𝑦𝑥
Distinct variable group:   𝑥,𝑦

Proof of Theorem axnulALT
Dummy variables 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ax-rep 4804 . . 3 (∀𝑤𝑥𝑦(∀𝑥⊥ → 𝑦 = 𝑥) → ∃𝑥𝑦(𝑦𝑥 ↔ ∃𝑤(𝑤𝑧 ∧ ∀𝑥⊥)))
2 sp 2091 . . . . . 6 (∀𝑥 ¬ ∀𝑦(∀𝑥⊥ → 𝑦 = 𝑥) → ¬ ∀𝑦(∀𝑥⊥ → 𝑦 = 𝑥))
32con2i 134 . . . . 5 (∀𝑦(∀𝑥⊥ → 𝑦 = 𝑥) → ¬ ∀𝑥 ¬ ∀𝑦(∀𝑥⊥ → 𝑦 = 𝑥))
4 df-ex 1745 . . . . 5 (∃𝑥𝑦(∀𝑥⊥ → 𝑦 = 𝑥) ↔ ¬ ∀𝑥 ¬ ∀𝑦(∀𝑥⊥ → 𝑦 = 𝑥))
53, 4sylibr 224 . . . 4 (∀𝑦(∀𝑥⊥ → 𝑦 = 𝑥) → ∃𝑥𝑦(∀𝑥⊥ → 𝑦 = 𝑥))
6 fal 1530 . . . . . 6 ¬ ⊥
7 sp 2091 . . . . . 6 (∀𝑥⊥ → ⊥)
86, 7mto 188 . . . . 5 ¬ ∀𝑥
98pm2.21i 116 . . . 4 (∀𝑥⊥ → 𝑦 = 𝑥)
105, 9mpg 1764 . . 3 𝑥𝑦(∀𝑥⊥ → 𝑦 = 𝑥)
111, 10mpg 1764 . 2 𝑥𝑦(𝑦𝑥 ↔ ∃𝑤(𝑤𝑧 ∧ ∀𝑥⊥))
128intnan 980 . . . . . 6 ¬ (𝑤𝑧 ∧ ∀𝑥⊥)
1312nex 1771 . . . . 5 ¬ ∃𝑤(𝑤𝑧 ∧ ∀𝑥⊥)
1413nbn 361 . . . 4 𝑦𝑥 ↔ (𝑦𝑥 ↔ ∃𝑤(𝑤𝑧 ∧ ∀𝑥⊥)))
1514albii 1787 . . 3 (∀𝑦 ¬ 𝑦𝑥 ↔ ∀𝑦(𝑦𝑥 ↔ ∃𝑤(𝑤𝑧 ∧ ∀𝑥⊥)))
1615exbii 1814 . 2 (∃𝑥𝑦 ¬ 𝑦𝑥 ↔ ∃𝑥𝑦(𝑦𝑥 ↔ ∃𝑤(𝑤𝑧 ∧ ∀𝑥⊥)))
1711, 16mpbir 221 1 𝑥𝑦 ¬ 𝑦𝑥
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  wal 1521   = wceq 1523  wfal 1528  wex 1744  wcel 2030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-12 2087  ax-rep 4804
This theorem depends on definitions:  df-bi 197  df-an 385  df-tru 1526  df-fal 1529  df-ex 1745
This theorem is referenced by: (None)
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