Theorem axc11 2444

Description: Show that ax-c11 34645 can be derived from ax-c11n 34646 in the form of axc11n 2439. Normally, axc11 2444 should be used rather than ax-c11 34645, except by theorems specifically studying the latter's properties. (Contributed by NM, 16-May-2008.) (Proof shortened by Wolf Lammen, 21-Apr-2018.)

Assertion

Ref	Expression
axc11	⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑))

Proof of Theorem axc11

Step	Hyp	Ref	Expression
1		axc11r 2320	. 2 ⊢ (∀𝑦 𝑦 = 𝑥 → (∀𝑥𝜑 → ∀𝑦𝜑))
2	1	aecoms 2442	1 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑))

Syntax hints:   → wi 4  ∀wal 1618

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-10 2156  ax-12 2184  ax-13 2379

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ex 1842  df-nf 1847

This theorem is referenced by:  hbae  2445  dral1  2453  dral1ALT  2454  nd1  9572  nd2  9573  axc11n11  32949  bj-hbaeb2  33082  wl-aetr  33599  ax6e2eq  39244  ax6e2eqVD  39611  2sb5ndVD  39614  2sb5ndALT  39636