Theorem axc11 2441

Description: Show that ax-c11 36183 can be derived from ax-c11n 36184 in the form of axc11n 2437. Normally, axc11 2441 should be used rather than ax-c11 36183, except by theorems specifically studying the latter's properties. Usage of this theorem is discouraged because it depends on ax-13 2379. Use the weaker axc11v 2262 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 2375	. 2 ⊢ (∀𝑦 𝑦 = 𝑥 → (∀𝑥𝜑 → ∀𝑦𝜑))
2	1	aecoms 2439	1 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑))

Syntax hints:   → wi 4  ∀wal 1536

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2142  ax-12 2175  ax-13 2379

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-nf 1786

This theorem is referenced by:  hbae  2442  dral1  2450  dral1ALT  2451  nd1  9998  nd2  9999  axc11n11  34129  bj-hbaeb2  34256  wl-aetr  34934  ax6e2eq  41263  ax6e2eqVD  41613  2sb5ndVD  41616  2sb5ndALT  41638