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Theorem a9evsep 3907
Description: Derive a weakened version of ax-i9 1439, where 𝑥 and 𝑦 must be distinct, from Separation ax-sep 3903 and Extensionality ax-ext 2038. The theorem ¬ ∀𝑥¬ 𝑥 = 𝑦 also holds (ax9vsep 3908), 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 3903 . 2 𝑥𝑧(𝑧𝑥 ↔ (𝑧𝑦 ∧ (𝑧 = 𝑧𝑧 = 𝑧)))
2 id 19 . . . . . . . 8 (𝑧 = 𝑧𝑧 = 𝑧)
32biantru 290 . . . . . . 7 (𝑧𝑦 ↔ (𝑧𝑦 ∧ (𝑧 = 𝑧𝑧 = 𝑧)))
43bibi2i 220 . . . . . 6 ((𝑧𝑥𝑧𝑦) ↔ (𝑧𝑥 ↔ (𝑧𝑦 ∧ (𝑧 = 𝑧𝑧 = 𝑧))))
54biimpri 128 . . . . 5 ((𝑧𝑥 ↔ (𝑧𝑦 ∧ (𝑧 = 𝑧𝑧 = 𝑧))) → (𝑧𝑥𝑧𝑦))
65alimi 1360 . . . 4 (∀𝑧(𝑧𝑥 ↔ (𝑧𝑦 ∧ (𝑧 = 𝑧𝑧 = 𝑧))) → ∀𝑧(𝑧𝑥𝑧𝑦))
7 ax-ext 2038 . . . 4 (∀𝑧(𝑧𝑥𝑧𝑦) → 𝑥 = 𝑦)
86, 7syl 14 . . 3 (∀𝑧(𝑧𝑥 ↔ (𝑧𝑦 ∧ (𝑧 = 𝑧𝑧 = 𝑧))) → 𝑥 = 𝑦)
98eximi 1507 . 2 (∃𝑥𝑧(𝑧𝑥 ↔ (𝑧𝑦 ∧ (𝑧 = 𝑧𝑧 = 𝑧))) → ∃𝑥 𝑥 = 𝑦)
101, 9ax-mp 7 1 𝑥 𝑥 = 𝑦
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  wal 1257   = wceq 1259  wex 1397  wcel 1409
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-4 1416  ax-ial 1443  ax-ext 2038  ax-sep 3903
This theorem depends on definitions:  df-bi 114
This theorem is referenced by:  ax9vsep  3908
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