Theorem axc11 2438

Description: Show that ax-c11 39386 can be derived from ax-c11n 39387 in the form of axc11n 2434. Normally, axc11 2438 should be used rather than ax-c11 39386, except by theorems specifically studying the latter's properties. Usage of this theorem is discouraged because it depends on ax-13 2380. Use the weaker axc11v 2276 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 2376	. 2 ⊢ (∀𝑦 𝑦 = 𝑥 → (∀𝑥𝜑 → ∀𝑦𝜑))
2	1	aecoms 2436	1 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑))

Syntax hints:   → wi 4  ∀wal 1545

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-10 2152  ax-12 2189  ax-13 2380

This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1787  df-nf 1791

This theorem is referenced by:  hbae  2439  dral1  2447  dral1ALT  2448  nd1  10508  nd2  10509  axsepg2  35328  axsepg4  35331  axc11n11  37032  bj-hbaeb2  37178  wl-aetr  37907  ax6e2eq  45008  ax6e2eqVD  45357  2sb5ndVD  45360  2sb5ndALT  45382