Theorem axc11 2452

Description: Show that ax-c11 36025 can be derived from ax-c11n 36026 in the form of axc11n 2448. Normally, axc11 2452 should be used rather than ax-c11 36025, except by theorems specifically studying the latter's properties. Usage of this theorem is discouraged because it depends on ax-13 2390. Use the weaker axc11v 2265 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 2386	. 2 ⊢ (∀𝑦 𝑦 = 𝑥 → (∀𝑥𝜑 → ∀𝑦𝜑))
2	1	aecoms 2450	1 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑))

Syntax hints:   → wi 4  ∀wal 1535

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 1911  ax-6 1970  ax-7 2015  ax-10 2145  ax-12 2177  ax-13 2390

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781  df-nf 1785

This theorem is referenced by:  hbae  2453  dral1  2461  dral1ALT  2462  nd1  10011  nd2  10012  axc11n11  34018  bj-hbaeb2  34143  wl-aetr  34771  ax6e2eq  40898  ax6e2eqVD  41248  2sb5ndVD  41251  2sb5ndALT  41273