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Theorem bj-ax89 31854
 Description: A theorem which could be used as sole axiom for the non-logical predicate instead of ax-8 1979 and ax-9 1986. Indeed, it is implied over propositional calculus by the conjunction of ax-8 1979 and ax-9 1986, as proved here. In the other direction, one can prove ax-8 1979 (respectively ax-9 1986) from bj-ax89 31854 by using mpan2 703 ( respectively mpan 702) and equid 1926. (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 1983 . 2 (𝑥 = 𝑦 → (𝑥𝑧𝑦𝑧))
2 ax9 1990 . 2 (𝑧 = 𝑡 → (𝑦𝑧𝑦𝑡))
31, 2sylan9 687 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 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986 This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696 This theorem is referenced by: (None)
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