Theorem bnj1212 30870

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

Hypotheses

Ref	Expression
bnj1212.1	|- B = { x e. A | ph }
bnj1212.2	|- ( th <-> ( ch /\ x e. B /\ ta ) )

Assertion

Ref	Expression
bnj1212	|- ( th -> x e. A )

Distinct variable group:   x, A

Allowed substitution hints:   ph( x)   ch( x)   th( x)   ta( x)   B( x)

Proof of Theorem bnj1212

Step	Hyp	Ref	Expression
1		bnj1212.1	. . 3 |- B = { x e. A | ph }
2	1	ssrab3 3688	. 2 |- B C_ A
3		bnj1212.2	. . 3 |- ( th <-> ( ch /\ x e. B /\ ta ) )
4	3	simp2bi 1077	. 2 |- ( th -> x e. B )
5	2, 4	bnj1213 30869	1 |- ( th -> x e. A )

Syntax hints:   -> wi 4   <-> wb 196   /\ w3a 1037   = wceq 1483   e. wcel 1990   {crab 2916

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-rab 2921  df-in 3581  df-ss 3588

This theorem is referenced by:  bnj1204  31080  bnj1296  31089  bnj1415  31106  bnj1421  31110  bnj1442  31117  bnj1452  31120  bnj1489  31124