Theorem inab 2264

Description: Intersection of two class abstractions.

Assertion

Ref	Expression
inab	|- ({x | ph} i^i {x | ps}) = {x | (ph /\ ps)}

Proof of Theorem inab

Step	Hyp	Ref	Expression
1		df-clab 1462	. . . . 5 |- (y e. {x | ph} <-> [y / x]ph)
2		df-clab 1462	. . . . 5 |- (y e. {x | ps} <-> [y / x]ps)
3	1, 2	anbi12i 482	. . . 4 |- ((y e. {x | ph} /\ y e. {x | ps}) <-> ([y / x]ph /\ [y / x]ps))
4		sban 1235	. . . 4 |- ([y / x](ph /\ ps) <-> ([y / x]ph /\ [y / x]ps))
5	3, 4	bitr4 176	. . 3 |- ((y e. {x | ph} /\ y e. {x | ps}) <-> [y / x](ph /\ ps))
6		elin 2203	. . 3 |- (y e. ({x | ph} i^i {x | ps}) <-> (y e. {x | ph} /\ y e. {x | ps}))
7		df-clab 1462	. . 3 |- (y e. {x | (ph /\ ps)} <-> [y / x](ph /\ ps))
8	5, 6, 7	3bitr4 183	. 2 |- (y e. ({x | ph} i^i {x | ps}) <-> y e. {x | (ph /\ ps)})
9	8	eqriv 1472	1 |- ({x | ph} i^i {x | ps}) = {x | (ph /\ ps)}

Syntax hints:   /\ wa 223   = wceq 954   e. wcel 956  [wsbc 1168  {cab 1461   i^i cin 2042

This theorem is referenced by:  difab 2265  inrab 2267  inrab2 2268  dfrab2 2270  orduniss2 3085  ssenen 4490  h2hcau 8788

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-v 1808  df-in 2047

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