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Theorem axext 206
Description: Axiom of Extensionality. An axiom of Zermelo-Fraenkel set theory. It states that two sets are identical if they contain the same elements. Axiom Ext of [BellMachover] p. 461.
Hypotheses
Ref Expression
axext.1 |- A:(al -> *)
axext.2 |- B:(al -> *)
Assertion
Ref Expression
axext |- T. |= [(A.\x:al [(Ax:al) = (Bx:al)]) ==> [A = B]]
Distinct variable groups:   x,A   x,B   al,x

Proof of Theorem axext
StepHypRef Expression
1 axext.1 . . . . . 6 |- A:(al -> *)
2 wv 58 . . . . . 6 |- x:al:al
31, 2wc 45 . . . . 5 |- (Ax:al):*
43wl 59 . . . 4 |- \x:al (Ax:al):(al -> *)
5 axext.2 . . . . . . . 8 |- B:(al -> *)
65, 2wc 45 . . . . . . 7 |- (Bx:al):*
73, 6weqi 68 . . . . . 6 |- [(Ax:al) = (Bx:al)]:*
87ax4 140 . . . . 5 |- (A.\x:al [(Ax:al) = (Bx:al)]) |= [(Ax:al) = (Bx:al)]
9 wal 124 . . . . . 6 |- A.:((al -> *) -> *)
107wl 59 . . . . . 6 |- \x:al [(Ax:al) = (Bx:al)]:(al -> *)
11 wv 58 . . . . . 6 |- y:al:al
129, 11ax-17 95 . . . . . 6 |- T. |= [(\x:al A.y:al) = A.]
137, 11ax-hbl1 93 . . . . . 6 |- T. |= [(\x:al \x:al [(Ax:al) = (Bx:al)]y:al) = \x:al [(Ax:al) = (Bx:al)]]
149, 10, 11, 12, 13hbc 100 . . . . 5 |- T. |= [(\x:al (A.\x:al [(Ax:al) = (Bx:al)])y:al) = (A.\x:al [(Ax:al) = (Bx:al)])]
153, 8, 14leqf 169 . . . 4 |- (A.\x:al [(Ax:al) = (Bx:al)]) |= [\x:al (Ax:al) = \x:al (Bx:al)]
1615ax-cb1 29 . . . . 5 |- (A.\x:al [(Ax:al) = (Bx:al)]):*
171eta 166 . . . . 5 |- T. |= [\x:al (Ax:al) = A]
1816, 17a1i 28 . . . 4 |- (A.\x:al [(Ax:al) = (Bx:al)]) |= [\x:al (Ax:al) = A]
195eta 166 . . . . 5 |- T. |= [\x:al (Bx:al) = B]
2016, 19a1i 28 . . . 4 |- (A.\x:al [(Ax:al) = (Bx:al)]) |= [\x:al (Bx:al) = B]
214, 15, 18, 203eqtr3i 87 . . 3 |- (A.\x:al [(Ax:al) = (Bx:al)]) |= [A = B]
22 wtru 40 . . 3 |- T.:*
2321, 22adantl 51 . 2 |- (T., (A.\x:al [(Ax:al) = (Bx:al)])) |= [A = B]
2423ex 148 1 |- T. |= [(A.\x:al [(Ax:al) = (Bx:al)]) ==> [A = B]]
Colors of variables: type var term
Syntax hints:  tv 1   -> ht 2  *hb 3  kc 5  \kl 6   = ke 7  T.kt 8  [kbr 9   |= wffMMJ2 11  wffMMJ2t 12   ==> tim 111  A.tal 112
This theorem was proved from axioms:  ax-syl 15  ax-jca 17  ax-simpl 20  ax-simpr 21  ax-id 24  ax-trud 26  ax-cb1 29  ax-cb2 30  ax-refl 39  ax-eqmp 42  ax-ded 43  ax-ceq 46  ax-beta 60  ax-distrc 61  ax-leq 62  ax-distrl 63  ax-hbl1 93  ax-17 95  ax-inst 103  ax-eta 165
This theorem depends on definitions:  df-ov 65  df-al 116  df-an 118  df-im 119
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