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Theorem bj-ax89 32792
 Description: A theorem which could be used as sole axiom for the non-logical predicate instead of ax-8 2032 and ax-9 2039. Indeed, it is implied over propositional calculus by the conjunction of ax-8 2032 and ax-9 2039, as proved here. In the other direction, one can prove ax-8 2032 (respectively ax-9 2039) from bj-ax89 32792 by using mpan2 707 ( respectively mpan 706) and equid 1985. (TODO: move to main part.) (Contributed by BJ, 3-Oct-2019.)
Assertion
Ref Expression
bj-ax89 ((𝑥 = 𝑦𝑧 = 𝑡) → (𝑥𝑧𝑦𝑡))

Proof of Theorem bj-ax89
StepHypRef Expression
1 ax8 2036 . 2 (𝑥 = 𝑦 → (𝑥𝑧𝑦𝑧))
2 ax9 2043 . 2 (𝑧 = 𝑡 → (𝑦𝑧𝑦𝑡))
31, 2sylan9 690 1 ((𝑥 = 𝑦𝑧 = 𝑡) → (𝑥𝑧𝑦𝑡))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383 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  ax-8 2032  ax-9 2039 This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1745 This theorem is referenced by: (None)
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