Theorem axc11 2430

Description: Show that ax-c11 38925 can be derived from ax-c11n 38926 in the form of axc11n 2426. Normally, axc11 2430 should be used rather than ax-c11 38925, except by theorems specifically studying the latter's properties. Usage of this theorem is discouraged because it depends on ax-13 2372. Use the weaker axc11v 2267 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 2368	. 2 ⊢ (∀𝑦 𝑦 = 𝑥 → (∀𝑥𝜑 → ∀𝑦𝜑))
2	1	aecoms 2428	1 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑))

Syntax hints:   → wi 4  ∀wal 1539

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-10 2144  ax-12 2180  ax-13 2372

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781  df-nf 1785

This theorem is referenced by:  hbae  2431  dral1  2439  dral1ALT  2440  nd1  10475  nd2  10476  axc11n11  36715  bj-hbaeb2  36851  wl-aetr  37562  ax6e2eq  44589  ax6e2eqVD  44938  2sb5ndVD  44941  2sb5ndALT  44963