Theorem axc11 2460

Description: Show that ax-c11 39475 can be derived from ax-c11n 39476 in the form of axc11n 2456. Normally, axc11 2460 should be used rather than ax-c11 39475, except by theorems specifically studying the latter's properties. Usage of this theorem is discouraged because it depends on ax-13 2402. Use the weaker axc11v 2298 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 2398	. 2 ⊢ (∀𝑦 𝑦 = 𝑥 → (∀𝑥𝜑 → ∀𝑦𝜑))
2	1	aecoms 2458	1 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑))

Syntax hints:   → wi 4  ∀wal 1557

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-10 2174  ax-12 2211  ax-13 2402

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1799  df-nf 1803

This theorem is referenced by:  hbae  2461  dral1  2469  dral1ALT  2470  nd1  10542  nd2  10543  axsepg2  35400  axsepg4  35403  axc11n11  37121  bj-hbaeb2  37267  wl-aetr  37996  ax6e2eq  45097  ax6e2eqVD  45446  2sb5ndVD  45449  2sb5ndALT  45471