Theorem axc11 2459

Description: Show that ax-c11 40038 can be derived from ax-c11n 40039 in the form of axc11n 2455. Normally, axc11 2459 should be used rather than ax-c11 40038, except by theorems specifically studying the latter's properties. Usage of this theorem is discouraged because it depends on ax-13 2401. Use the weaker axc11v 2298 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 2397	. 2 ⊢ (∀𝑦 𝑦 = 𝑥 → (∀𝑥𝜑 → ∀𝑦𝜑))
2	1	aecoms 2457	1 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑))

This proof depends on syntax axioms:   → wi 4  ∀wal 1567

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-10 2177  ax-12 2212  ax-13 2401

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-nf 1816

This theorem is used by:  hbae  2460  dral1  2468  dral1ALT  2469  nd1  10676  nd2  10677  axsepg2  35865  axsepg4  35868  axc11n11  37677  bj-hbaeb2  37823  wl-aetr  38555  ax6e2eq  45634  ax6e2eqVD  45983  2sb5ndVD  45986  2sb5ndALT  46008