Theorem bj-elisset 32862

Description: Remove from elisset 3215 dependency on ax-ext 2602 (and on df-cleq 2615 and df-v 3202). This proof uses only df-clab 2609 and df-clel 2618 on top of first-order logic. It only requires ax-1--7 and sp 2053. Use bj-elissetv 32861 instead when sufficient (in particular when V is substituted for _V). (Contributed by BJ, 29-Apr-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-elisset	|- ( A e. V -> E. x x = A )

Distinct variable group:   x, A

Allowed substitution hint:   V( x)

Proof of Theorem bj-elisset

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bj-elissetv 32861	. 2 |- ( A e. V -> E. y y = A )
2		bj-denotes 32858	. 2 |- ( E. y y = A <-> E. x x = A )
3	1, 2	sylib 208	1 |- ( A e. V -> E. x x = A )

Syntax hints:   -> wi 4   = wceq 1483   E.wex 1704   e. wcel 1990

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-sb 1881  df-clab 2609  df-clel 2618

This theorem is referenced by:  bj-isseti  32864  bj-ceqsalt  32875  bj-ceqsalg  32878  bj-spcimdv  32884  bj-vtoclg1f  32911