Theorem axc16g-o 34219

Description: A generalization of axiom ax-c16 34177. Version of axc16g 2134 using ax-c11 34172. (Contributed by NM, 15-May-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
axc16g-o	|- ( A. x x = y -> ( ph -> A. z ph ) )

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y, z)

Proof of Theorem axc16g-o

Step	Hyp	Ref	Expression
1		aev-o 34216	. 2 |- ( A. x x = y -> A. z z = x )
2		ax-c16 34177	. 2 |- ( A. x x = y -> ( ph -> A. x ph ) )
3		biidd 252	. . . 4 |- ( A. z z = x -> ( ph <-> ph ) )
4	3	dral1-o 34189	. . 3 |- ( A. z z = x -> ( A. z ph <-> A. x ph ) )
5	4	biimprd 238	. 2 |- ( A. z z = x -> ( A. x ph -> A. z ph ) )
6	1, 2, 5	sylsyld 61	1 |- ( A. x x = y -> ( ph -> A. z ph ) )

Syntax hints:   -> wi 4   A.wal 1481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-11 2034  ax-12 2047  ax-13 2246  ax-c5 34168  ax-c4 34169  ax-c7 34170  ax-c10 34171  ax-c11 34172  ax-c9 34175  ax-c16 34177

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705

This theorem is referenced by:  ax12inda2  34232