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Theorem axc11 2453
Description: Show that ax-c11 36141 can be derived from ax-c11n 36142 in the form of axc11n 2449. Normally, axc11 2453 should be used rather than ax-c11 36141, except by theorems specifically studying the latter's properties. Usage of this theorem is discouraged because it depends on ax-13 2391. Use the weaker axc11v 2266 when possible. (Contributed by NM, 16-May-2008.) (Proof shortened by Wolf Lammen, 21-Apr-2018.) (New usage is discouraged.)
Assertion
Ref Expression
axc11 (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑))

Proof of Theorem axc11
StepHypRef Expression
1 axc11r 2387 . 2 (∀𝑦 𝑦 = 𝑥 → (∀𝑥𝜑 → ∀𝑦𝜑))
21aecoms 2451 1 (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  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 1911  ax-6 1970  ax-7 2015  ax-10 2145  ax-12 2178  ax-13 2391
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-nf 1786
This theorem is referenced by:  hbae  2454  dral1  2462  dral1ALT  2463  nd1  9998  nd2  9999  axc11n11  34090  bj-hbaeb2  34217  wl-aetr  34892  ax6e2eq  41197  ax6e2eqVD  41547  2sb5ndVD  41550  2sb5ndALT  41572
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