Theorem axc11 2460

Description: Show that ax-c11 39912 can be derived from ax-c11n 39913 in the form of axc11n 2456. Normally, axc11 2460 should be used rather than ax-c11 39912, except by theorems specifically studying the latter's properties. Usage of this theorem is discouraged because it depends on ax-13 2402. Use the weaker axc11v 2299 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

Step	Hyp	Ref	Expression
1		axc11r 2398	. 2 ⊢ (∀𝑦 𝑦 = 𝑥 → (∀𝑥𝜑 → ∀𝑦𝜑))
2	1	aecoms 2458	1 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑))

This proof depends on syntax axioms:   → wi 4  ∀wal 1568

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-10 2178  ax-12 2213  ax-13 2402

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-nf 1817

This theorem is used by:  hbae  2461  dral1  2469  dral1ALT  2470  nd1  10653  nd2  10654  axsepg2  35781  axsepg4  35784  axc11n11  37554  bj-hbaeb2  37700  wl-aetr  38429  ax6e2eq  45499  ax6e2eqVD  45848  2sb5ndVD  45851  2sb5ndALT  45873