Theorem bnj90 30788

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Mario Carneiro, 22-Dec-2016.) (New usage is discouraged.)

Hypothesis

Ref	Expression
bnj90.1	|- Y e. _V

Assertion

Ref	Expression
bnj90	|- ( [. Y / x ]. z Fn x <-> z Fn Y )

Distinct variable group:   x, z

Allowed substitution hints:   Y( x, z)

Proof of Theorem bnj90

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bnj90.1	. 2 |- Y e. _V
2		fneq2 5980	. . 3 |- ( x = y -> ( z Fn x <-> z Fn y ) )
3		fneq2 5980	. . 3 |- ( y = Y -> ( z Fn y <-> z Fn Y ) )
4	2, 3	sbcie2g 3469	. 2 |- ( Y e. _V -> ( [. Y / x ]. z Fn x <-> z Fn Y ) )
5	1, 4	ax-mp 5	1 |- ( [. Y / x ]. z Fn x <-> z Fn Y )

Syntax hints:   <-> wb 196   e. wcel 1990   _Vcvv 3200   [.wsbc 3435   Fn wfn 5883

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-9 1999  ax-10 2019  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-v 3202  df-sbc 3436  df-fn 5891

This theorem is referenced by:  bnj121  30940  bnj130  30944  bnj207  30951