Theorem axc11 2428

Description: Show that ax-c11 37147 can be derived from ax-c11n 37148 in the form of axc11n 2424. Normally, axc11 2428 should be used rather than ax-c11 37147, except by theorems specifically studying the latter's properties. Usage of this theorem is discouraged because it depends on ax-13 2370. Use the weaker axc11v 2255 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 2364	. 2 ⊢ (∀𝑦 𝑦 = 𝑥 → (∀𝑥𝜑 → ∀𝑦𝜑))
2	1	aecoms 2426	1 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑))

Syntax hints:   → wi 4  ∀wal 1538

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 1912  ax-6 1970  ax-7 2010  ax-10 2136  ax-12 2170  ax-13 2370

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1781  df-nf 1785

This theorem is referenced by:  hbae  2429  dral1  2437  dral1ALT  2438  nd1  10436  nd2  10437  axc11n11  34955  bj-hbaeb2  35091  wl-aetr  35786  ax6e2eq  42487  ax6e2eqVD  42837  2sb5ndVD  42840  2sb5ndALT  42862