Theorem inab 2268

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 1464	. . . . 5 |- (y e. {x | ph} <-> [y / x]ph)
2		df-clab 1464	. . . . 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 1237	. . . 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 2207	. . 3 |- (y e. ({x | ph} i^i {x | ps}) <-> (y e. {x | ph} /\ y e. {x | ps}))
7		df-clab 1464	. . 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 1474	1 |- ({x | ph} i^i {x | ps}) = {x | (ph /\ ps)}

Syntax hints:   /\ wa 223   = wceq 956   e. wcel 958  [wsbc 1170  {cab 1463   i^i cin 2046

This theorem is referenced by:  difab 2269  inrab 2271  inrab2 2272  dfrab2 2274  orduniss2 3090  ssenen 4504  h2hcau 8849

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-in 2051

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