Theorem axc11 2462

Description: Show that ax-c11 39642 can be derived from ax-c11n 39643 in the form of axc11n 2458. Normally, axc11 2462 should be used rather than ax-c11 39642, except by theorems specifically studying the latter's properties. Usage of this theorem is discouraged because it depends on ax-13 2404. Use the weaker axc11v 2300 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 2400	. 2 ⊢ (∀𝑦 𝑦 = 𝑥 → (∀𝑥𝜑 → ∀𝑦𝜑))
2	1	aecoms 2460	1 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑))

Syntax hints:   → wi 4  ∀wal 1568

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-10 2176  ax-12 2213  ax-13 2404

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1810  df-nf 1814

This theorem is referenced by:  hbae  2463  dral1  2471  dral1ALT  2472  nd1  10573  nd2  10574  axsepg2  35534  axsepg4  35537  axc11n11  37288  bj-hbaeb2  37434  wl-aetr  38165  ax6e2eq  45249  ax6e2eqVD  45598  2sb5ndVD  45601  2sb5ndALT  45623