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Theorem equviniOLD 2474
Description: Obsolete version of equvini 2473 as of 16-Sep-2023. (Contributed by NM, 10-Jan-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 15-Sep-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
equviniOLD (𝑥 = 𝑦 → ∃𝑧(𝑥 = 𝑧𝑧 = 𝑦))

Proof of Theorem equviniOLD
StepHypRef Expression
1 equtr 2024 . . . 4 (𝑧 = 𝑥 → (𝑥 = 𝑦𝑧 = 𝑦))
2 equeuclr 2026 . . . . 5 (𝑧 = 𝑦 → (𝑥 = 𝑦𝑥 = 𝑧))
32anc2ri 559 . . . 4 (𝑧 = 𝑦 → (𝑥 = 𝑦 → (𝑥 = 𝑧𝑧 = 𝑦)))
41, 3syli 39 . . 3 (𝑧 = 𝑥 → (𝑥 = 𝑦 → (𝑥 = 𝑧𝑧 = 𝑦)))
5 19.8a 2176 . . 3 ((𝑥 = 𝑧𝑧 = 𝑦) → ∃𝑧(𝑥 = 𝑧𝑧 = 𝑦))
64, 5syl6 35 . 2 (𝑧 = 𝑥 → (𝑥 = 𝑦 → ∃𝑧(𝑥 = 𝑧𝑧 = 𝑦)))
7 ax13 2389 . . 3 𝑧 = 𝑥 → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
8 ax6e 2397 . . . . 5 𝑧 𝑧 = 𝑦
98, 3eximii 1833 . . . 4 𝑧(𝑥 = 𝑦 → (𝑥 = 𝑧𝑧 = 𝑦))
10919.35i 1875 . . 3 (∀𝑧 𝑥 = 𝑦 → ∃𝑧(𝑥 = 𝑧𝑧 = 𝑦))
117, 10syl6 35 . 2 𝑧 = 𝑥 → (𝑥 = 𝑦 → ∃𝑧(𝑥 = 𝑧𝑧 = 𝑦)))
126, 11pm2.61i 184 1 (𝑥 = 𝑦 → ∃𝑧(𝑥 = 𝑧𝑧 = 𝑦))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 398  wal 1531  wex 1776
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-12 2173  ax-13 2386
This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1777
This theorem is referenced by: (None)
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