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Theorem equvini 2333
Description: A variable introduction law for equality. Lemma 15 of [Monk2] p. 109, however we do not require 𝑧 to be distinct from 𝑥 and 𝑦. See equvinv 1945 for a shorter proof requiring fewer axioms when 𝑧 is required to be distinct from 𝑥 and 𝑦. (Contributed by NM, 10-Jan-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 15-Sep-2018.)
Assertion
Ref Expression
equvini (𝑥 = 𝑦 → ∃𝑧(𝑥 = 𝑧𝑧 = 𝑦))

Proof of Theorem equvini
StepHypRef Expression
1 equtr 1934 . . . 4 (𝑧 = 𝑥 → (𝑥 = 𝑦𝑧 = 𝑦))
2 equeuclr 1936 . . . . 5 (𝑧 = 𝑦 → (𝑥 = 𝑦𝑥 = 𝑧))
32anc2ri 578 . . . 4 (𝑧 = 𝑦 → (𝑥 = 𝑦 → (𝑥 = 𝑧𝑧 = 𝑦)))
41, 3syli 38 . . 3 (𝑧 = 𝑥 → (𝑥 = 𝑦 → (𝑥 = 𝑧𝑧 = 𝑦)))
5 19.8a 2038 . . 3 ((𝑥 = 𝑧𝑧 = 𝑦) → ∃𝑧(𝑥 = 𝑧𝑧 = 𝑦))
64, 5syl6 34 . 2 (𝑧 = 𝑥 → (𝑥 = 𝑦 → ∃𝑧(𝑥 = 𝑧𝑧 = 𝑦)))
7 ax13 2236 . . 3 𝑧 = 𝑥 → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
8 ax6e 2237 . . . . 5 𝑧 𝑧 = 𝑦
98, 3eximii 1753 . . . 4 𝑧(𝑥 = 𝑦 → (𝑥 = 𝑧𝑧 = 𝑦))
10919.35i 1794 . . 3 (∀𝑧 𝑥 = 𝑦 → ∃𝑧(𝑥 = 𝑧𝑧 = 𝑦))
117, 10syl6 34 . 2 𝑧 = 𝑥 → (𝑥 = 𝑦 → ∃𝑧(𝑥 = 𝑧𝑧 = 𝑦)))
126, 11pm2.61i 174 1 (𝑥 = 𝑦 → ∃𝑧(𝑥 = 𝑧𝑧 = 𝑦))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382  wal 1472  wex 1694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-12 2033  ax-13 2233
This theorem depends on definitions:  df-bi 195  df-an 384  df-ex 1695
This theorem is referenced by:  2ax6elem  2436
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