Theorem axc11 2462

Description: Show that ax-c11 40147 can be derived from ax-c11n 40148 in the form of axc11n 2458. Normally, axc11 2462 should be used rather than ax-c11 40147, except by theorems specifically studying the latter's properties. Usage of this theorem is discouraged because it depends on ax-13 2404. Use the weaker axc11v 2302 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 2400	. 2 ⊢ (∀𝑦 𝑦 = 𝑥 → (∀𝑥𝜑 → ∀𝑦𝜑))
2	1	aecoms 2460	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 1945  ax-6 2002  ax-7 2043  ax-10 2180  ax-12 2216  ax-13 2404

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

This theorem is used by:  hbae  2463  dral1  2471  dral1ALT  2472  nd1  10702  nd2  10703  axsepg2  35921  axsepg4  35924  axc11n11  37775  bj-hbaeb2  37921  wl-aetr  38666  ax6e2eq  45742  ax6e2eqVD  46091  2sb5ndVD  46094  2sb5ndALT  46116