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Theorem bj-ax89 33173
Description: A theorem which could be used as sole axiom for the non-logical predicate instead of ax-8 2159 and ax-9 2166. Indeed, it is implied over propositional calculus by the conjunction of ax-8 2159 and ax-9 2166, as proved here. In the other direction, one can prove ax-8 2159 (respectively ax-9 2166) from bj-ax89 33173 by using mpan2 683 ( respectively mpan 682) and equid 2111. (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 2163 . 2 (𝑥 = 𝑦 → (𝑥𝑧𝑦𝑧))
2 ax9 2170 . 2 (𝑧 = 𝑡 → (𝑦𝑧𝑦𝑡))
31, 2sylan9 504 1 ((𝑥 = 𝑦𝑧 = 𝑡) → (𝑥𝑧𝑦𝑡))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166
This theorem depends on definitions:  df-bi 199  df-an 386  df-ex 1876
This theorem is referenced by: (None)
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