Theorem axc11 2465

Description: Show that ax-c11 39702 can be derived from ax-c11n 39703 in the form of axc11n 2461. Normally, axc11 2465 should be used rather than ax-c11 39702, except by theorems specifically studying the latter's properties. Usage of this theorem is discouraged because it depends on ax-13 2407. Use the weaker axc11v 2303 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 2403	. 2 ⊢ (∀𝑦 𝑦 = 𝑥 → (∀𝑥𝜑 → ∀𝑦𝜑))
2	1	aecoms 2463	1 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑))

This proof depends on syntax axioms:   → wi 4  ∀wal 1568

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-10 2179  ax-12 2216  ax-13 2407

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-nf 1817

This theorem is used by:  hbae  2466  dral1  2474  dral1ALT  2475  nd1  10590  nd2  10591  axsepg2  35577  axsepg4  35580  axc11n11  37348  bj-hbaeb2  37494  wl-aetr  38225  ax6e2eq  45307  ax6e2eqVD  45656  2sb5ndVD  45659  2sb5ndALT  45681