Theorem axc11 2313

Description: Show that ax-c11 33649 can be derived from ax-c11n 33650 in the form of axc11n 2306. Normally, axc11 2313 should be used rather than ax-c11 33649, except by theorems specifically studying the latter's properties. (Contributed by NM, 16-May-2008.) (Proof shortened by Wolf Lammen, 21-Apr-2018.)

Assertion

Ref	Expression
axc11	⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑))

Proof of Theorem axc11

Step	Hyp	Ref	Expression
1		axc11r 2186	. 2 ⊢ (∀𝑦 𝑦 = 𝑥 → (∀𝑥𝜑 → ∀𝑦𝜑))
2	1	aecoms 2311	1 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑))

Syntax hints:   → wi 4  ∀wal 1478

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-10 2016  ax-12 2044  ax-13 2245

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1702  df-nf 1707

This theorem is referenced by:  hbae  2314  dral1  2324  dral1ALT  2325  nd1  9353  nd2  9354  axc11n11  32311  bj-hbaeb2  32445  wl-aetr  32946  ax6e2eq  38252  ax6e2eqVD  38623  2sb5ndVD  38626  2sb5ndALT  38648