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Theorem bj-ax6elem2 32777
Description: Lemma for bj-ax6e 32778. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-ax6elem2 (∀𝑥 𝑦 = 𝑧 → ∃𝑥 𝑥 = 𝑦)
Distinct variable group:   𝑥,𝑧

Proof of Theorem bj-ax6elem2
StepHypRef Expression
1 ax6ev 1947 . . 3 𝑥 𝑥 = 𝑧
2 equeucl 1997 . . 3 (𝑥 = 𝑧 → (𝑦 = 𝑧𝑥 = 𝑦))
31, 2eximii 1804 . 2 𝑥(𝑦 = 𝑧𝑥 = 𝑦)
4319.35i 1846 1 (∀𝑥 𝑦 = 𝑧 → ∃𝑥 𝑥 = 𝑦)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1521  wex 1744
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
This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1745
This theorem is referenced by:  bj-ax6e  32778
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