Theorem axc11 2430

Description: Show that ax-c11 36828 can be derived from ax-c11n 36829 in the form of axc11n 2426. Normally, axc11 2430 should be used rather than ax-c11 36828, except by theorems specifically studying the latter's properties. Usage of this theorem is discouraged because it depends on ax-13 2372. Use the weaker axc11v 2259 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 2366	. 2 ⊢ (∀𝑦 𝑦 = 𝑥 → (∀𝑥𝜑 → ∀𝑦𝜑))
2	1	aecoms 2428	1 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑))

Syntax hints:   → wi 4  ∀wal 1537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-10 2139  ax-12 2173  ax-13 2372

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1784  df-nf 1788

This theorem is referenced by:  hbae  2431  dral1  2439  dral1ALT  2440  nd1  10274  nd2  10275  axc11n11  34791  bj-hbaeb2  34928  wl-aetr  35615  ax6e2eq  42066  ax6e2eqVD  42416  2sb5ndVD  42419  2sb5ndALT  42441