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Theorem a9evsep 4050
 Description: Derive a weakened version of ax-i9 1510, where 𝑥 and 𝑦 must be distinct, from Separation ax-sep 4046 and Extensionality ax-ext 2121. The theorem ¬ ∀𝑥¬ 𝑥 = 𝑦 also holds (ax9vsep 4051), but in intuitionistic logic ∃𝑥𝑥 = 𝑦 is stronger. (Contributed by Jim Kingdon, 25-Aug-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
a9evsep 𝑥 𝑥 = 𝑦
Distinct variable group:   𝑥,𝑦

Proof of Theorem a9evsep
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 ax-sep 4046 . 2 𝑥𝑧(𝑧𝑥 ↔ (𝑧𝑦 ∧ (𝑧 = 𝑧𝑧 = 𝑧)))
2 id 19 . . . . . . . 8 (𝑧 = 𝑧𝑧 = 𝑧)
32biantru 300 . . . . . . 7 (𝑧𝑦 ↔ (𝑧𝑦 ∧ (𝑧 = 𝑧𝑧 = 𝑧)))
43bibi2i 226 . . . . . 6 ((𝑧𝑥𝑧𝑦) ↔ (𝑧𝑥 ↔ (𝑧𝑦 ∧ (𝑧 = 𝑧𝑧 = 𝑧))))
54biimpri 132 . . . . 5 ((𝑧𝑥 ↔ (𝑧𝑦 ∧ (𝑧 = 𝑧𝑧 = 𝑧))) → (𝑧𝑥𝑧𝑦))
65alimi 1431 . . . 4 (∀𝑧(𝑧𝑥 ↔ (𝑧𝑦 ∧ (𝑧 = 𝑧𝑧 = 𝑧))) → ∀𝑧(𝑧𝑥𝑧𝑦))
7 ax-ext 2121 . . . 4 (∀𝑧(𝑧𝑥𝑧𝑦) → 𝑥 = 𝑦)
86, 7syl 14 . . 3 (∀𝑧(𝑧𝑥 ↔ (𝑧𝑦 ∧ (𝑧 = 𝑧𝑧 = 𝑧))) → 𝑥 = 𝑦)
98eximi 1579 . 2 (∃𝑥𝑧(𝑧𝑥 ↔ (𝑧𝑦 ∧ (𝑧 = 𝑧𝑧 = 𝑧))) → ∃𝑥 𝑥 = 𝑦)
101, 9ax-mp 5 1 𝑥 𝑥 = 𝑦
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  ∀wal 1329   = wceq 1331  ∃wex 1468   ∈ wcel 1480 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-ial 1514  ax-ext 2121  ax-sep 4046 This theorem depends on definitions:  df-bi 116 This theorem is referenced by:  ax9vsep  4051
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