Theorem bj-ax9 32890

Description: Proof of ax-9 1999 from Tarski's FOL=, sp 2053, df-cleq 2615 and ax-ext 2602 (with two extra dv conditions on x , z and y , z). For a version without these dv conditions, see bj-ax9-2 32891. This shows that df-cleq 2615 is "too powerful". A possible definition is given by bj-df-cleq 32893. (Contributed by BJ, 24-Jun-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-ax9	|- ( x = y -> ( z e. x -> z e. y ) )

Distinct variable groups:   x, z   y, z

Proof of Theorem bj-ax9

Dummy variables w u are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-ext 2602	. . 3 |- ( A. z ( z e. u <-> z e. w ) -> u = w )
2	1	df-cleq 2615	. 2 |- ( x = y <-> A. z ( z e. x <-> z e. y ) )
3		biimp 205	. . 3 |- ( ( z e. x <-> z e. y ) -> ( z e. x -> z e. y ) )
4	3	sps 2055	. 2 |- ( A. z ( z e. x <-> z e. y ) -> ( z e. x -> z e. y ) )
5	2, 4	sylbi 207	1 |- ( x = y -> ( z e. x -> z e. y ) )

Syntax hints:   -> wi 4   <-> wb 196   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-12 2047  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-ex 1705  df-cleq 2615

This theorem is referenced by: (None)