Theorem bnj538 30809

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) (Proof shortened by OpenAI, 30-Mar-2020.)

Hypothesis

Ref	Expression
bnj538.1	|- A e. _V

Assertion

Ref	Expression
bnj538	|- ( [. A / y ]. A. x e. B ph <-> A. x e. B [. A / y ]. ph )

Distinct variable groups:   x, A   y, B   x, y

Allowed substitution hints:   ph( x, y)   A( y)   B( x)

Proof of Theorem bnj538

Step	Hyp	Ref	Expression
1		bnj538.1	. 2 |- A e. _V
2		sbcralg 3513	. 2 |- ( A e. _V -> ( [. A / y ]. A. x e. B ph <-> A. x e. B [. A / y ]. ph ) )
3	1, 2	ax-mp 5	1 |- ( [. A / y ]. A. x e. B ph <-> A. x e. B [. A / y ]. ph )

Syntax hints:   <-> wb 196   e. wcel 1990   A.wral 2912   _Vcvv 3200   [.wsbc 3435

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-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-v 3202  df-sbc 3436

This theorem is referenced by:  bnj92  30932  bnj539  30961  bnj540  30962