Theorem bnj25OLD 12393

Description: First-order logic and set theory.

Hypothesis

Ref	Expression
bnj25.1	|- A = B

Assertion

Ref	Expression
bnj25OLD	|- (w e/ A <-> w e/ B)

Proof of Theorem bnj25OLD

Step	Hyp	Ref	Expression
1		bnj25.1	. . . 4 |- A = B
2	1	eleq2i 1961	. . 3 |- (w e. A <-> w e. B)
3	2	notbii 204	. 2 |- (-. w e. A <-> -. w e. B)
4		df-nel 2020	. 2 |- (w e/ A <-> -. w e. A)
5		df-nel 2020	. 2 |- (w e/ B <-> -. w e. B)
6	3, 4, 5	3bitr4i 200	1 |- (w e/ A <-> w e/ B)

Syntax hints:  -. wn 2   <-> wb 163   = wceq 1298   e. wcel 1300   e/ wnel 2018

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 1305  ax-17 1317  ax-4 1319  ax-5o 1321  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-an 242  df-ex 1327  df-cleq 1877  df-clel 1880  df-nel 2020

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