Theorem axc11 2464

Description: Show that ax-c11 39523 can be derived from ax-c11n 39524 in the form of axc11n 2460. Normally, axc11 2464 should be used rather than ax-c11 39523, except by theorems specifically studying the latter's properties. Usage of this theorem is discouraged because it depends on ax-13 2406. Use the weaker axc11v 2302 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 2402	. 2 ⊢ (∀𝑦 𝑦 = 𝑥 → (∀𝑥𝜑 → ∀𝑦𝜑))
2	1	aecoms 2462	1 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑))

Syntax hints:   → wi 4  ∀wal 1561

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-10 2178  ax-12 2215  ax-13 2406

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1803  df-nf 1807

This theorem is referenced by:  hbae  2465  dral1  2473  dral1ALT  2474  nd1  10560  nd2  10561  axsepg2  35448  axsepg4  35451  axc11n11  37169  bj-hbaeb2  37315  wl-aetr  38044  ax6e2eq  45131  ax6e2eqVD  45480  2sb5ndVD  45483  2sb5ndALT  45505