Theorem axc11 2434

Description: Show that ax-c11 39143 can be derived from ax-c11n 39144 in the form of axc11n 2430. Normally, axc11 2434 should be used rather than ax-c11 39143, except by theorems specifically studying the latter's properties. Usage of this theorem is discouraged because it depends on ax-13 2376. Use the weaker axc11v 2271 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 2372	. 2 ⊢ (∀𝑦 𝑦 = 𝑥 → (∀𝑥𝜑 → ∀𝑦𝜑))
2	1	aecoms 2432	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 2146  ax-12 2184  ax-13 2376

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

This theorem is referenced by:  hbae  2435  dral1  2443  dral1ALT  2444  nd1  10498  nd2  10499  axc11n11  36883  bj-hbaeb2  37019  wl-aetr  37730  ax6e2eq  44794  ax6e2eqVD  45143  2sb5ndVD  45146  2sb5ndALT  45168