Theorem axc16nf 2137

Description: If dtru 4857 is false, then there is only one element in the universe, so everything satisfies F/. (Contributed by Mario Carneiro, 7-Oct-2016.) Remove dependency on ax-11 2034. (Revised by Wolf Lammen, 9-Sep-2018.) (Proof shortened by BJ, 14-Jun-2019.) Remove dependency on ax-10 2019. (Revised by Wolf lammen, 12-Oct-2021.)

Assertion

Ref	Expression
axc16nf	|- ( A. x x = y -> F/ z ph )

Distinct variable group:   x, y

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

Proof of Theorem axc16nf

Step	Hyp	Ref	Expression
1		df-ex 1705	. . . 4 |- ( E. z ph <-> -. A. z -. ph )
2		axc16g 2134	. . . . 5 |- ( A. x x = y -> ( -. ph -> A. z -. ph ) )
3	2	con1d 139	. . . 4 |- ( A. x x = y -> ( -. A. z -. ph -> ph ) )
4	1, 3	syl5bi 232	. . 3 |- ( A. x x = y -> ( E. z ph -> ph ) )
5		axc16g 2134	. . 3 |- ( A. x x = y -> ( ph -> A. z ph ) )
6	4, 5	syld 47	. 2 |- ( A. x x = y -> ( E. z ph -> A. z ph ) )
7	6	nfd 1716	1 |- ( A. x x = y -> F/ z ph )

Syntax hints:   -. wn 3   -> wi 4   A.wal 1481   E.wex 1704   F/wnf 1708

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-12 2047

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-nf 1710

This theorem is referenced by:  nfsb  2440  nfsbd  2442