Mathbox for BJ < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-ax9 Structured version   Visualization version   GIF version

Theorem bj-ax9 34284
 Description: Proof of ax-9 2125 from Tarski's FOL=, sp 2184, dfcleq 2818 and ax-ext 2796 (with two extra disjoint variable conditions on 𝑥, 𝑧 and 𝑦, 𝑧). See ax9ALT 2820 for a more general version, proved using also ax-8 2117. (Contributed by BJ, 24-Jun-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
bj-ax9 (𝑥 = 𝑦 → (𝑧𝑥𝑧𝑦))
Distinct variable groups:   𝑥,𝑧   𝑦,𝑧

Proof of Theorem bj-ax9
StepHypRef Expression
1 dfcleq 2818 . 2 (𝑥 = 𝑦 ↔ ∀𝑧(𝑧𝑥𝑧𝑦))
2 biimp 218 . . 3 ((𝑧𝑥𝑧𝑦) → (𝑧𝑥𝑧𝑦))
32sps 2186 . 2 (∀𝑧(𝑧𝑥𝑧𝑦) → (𝑧𝑥𝑧𝑦))
41, 3sylbi 220 1 (𝑥 = 𝑦 → (𝑧𝑥𝑧𝑦))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1536 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-9 2125  ax-12 2179  ax-ext 2796 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-cleq 2817 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator