Theorem axc11 2435

Description: Show that ax-c11 39257 can be derived from ax-c11n 39258 in the form of axc11n 2431. Normally, axc11 2435 should be used rather than ax-c11 39257, except by theorems specifically studying the latter's properties. Usage of this theorem is discouraged because it depends on ax-13 2377. Use the weaker axc11v 2272 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 2373	. 2 ⊢ (∀𝑦 𝑦 = 𝑥 → (∀𝑥𝜑 → ∀𝑦𝜑))
2	1	aecoms 2433	1 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑))

Syntax hints:   → wi 4  ∀wal 1540

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 1912  ax-6 1969  ax-7 2010  ax-10 2147  ax-12 2185  ax-13 2377

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-nf 1786

This theorem is referenced by:  hbae  2436  dral1  2444  dral1ALT  2445  nd1  10510  nd2  10511  axc11n11  36921  bj-hbaeb2  37060  wl-aetr  37778  ax6e2eq  44907  ax6e2eqVD  45256  2sb5ndVD  45259  2sb5ndALT  45281