Theorem axc11 2429

Description: Show that ax-c11 38887 can be derived from ax-c11n 38888 in the form of axc11n 2425. Normally, axc11 2429 should be used rather than ax-c11 38887, except by theorems specifically studying the latter's properties. Usage of this theorem is discouraged because it depends on ax-13 2371. Use the weaker axc11v 2265 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 2367	. 2 ⊢ (∀𝑦 𝑦 = 𝑥 → (∀𝑥𝜑 → ∀𝑦𝜑))
2	1	aecoms 2427	1 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑))

Syntax hints:   → wi 4  ∀wal 1538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-10 2142  ax-12 2178  ax-13 2371

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-nf 1784

This theorem is referenced by:  hbae  2430  dral1  2438  dral1ALT  2439  nd1  10547  nd2  10548  axc11n11  36677  bj-hbaeb2  36813  wl-aetr  37524  ax6e2eq  44554  ax6e2eqVD  44903  2sb5ndVD  44906  2sb5ndALT  44928