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Theorem ax2 191
Description: Axiom Frege. Axiom A2 of [Margaris] p. 49.
Hypotheses
Ref Expression
ax1.1 |- R:*
ax1.2 |- S:*
ax2.3 |- T:*
Assertion
Ref Expression
ax2 |- T. |= [[R ==> [S ==> T]] ==> [[R ==> S] ==> [R ==> T]]]

Proof of Theorem ax2
StepHypRef Expression
1 ax2.3 . . . . . 6 |- T:*
2 ax1.2 . . . . . . 7 |- S:*
3 wim 127 . . . . . . . . . 10 |- ==> :(* -> (* -> *))
4 ax1.1 . . . . . . . . . 10 |- R:*
53, 2, 1wov 64 . . . . . . . . . 10 |- [S ==> T]:*
63, 4, 5wov 64 . . . . . . . . 9 |- [R ==> [S ==> T]]:*
73, 4, 2wov 64 . . . . . . . . 9 |- [R ==> S]:*
86, 7wct 44 . . . . . . . 8 |- ([R ==> [S ==> T]], [R ==> S]):*
98, 4simpr 23 . . . . . . 7 |- (([R ==> [S ==> T]], [R ==> S]), R) |= R
108, 4simpl 22 . . . . . . . 8 |- (([R ==> [S ==> T]], [R ==> S]), R) |= ([R ==> [S ==> T]], [R ==> S])
1110simprd 36 . . . . . . 7 |- (([R ==> [S ==> T]], [R ==> S]), R) |= [R ==> S]
122, 9, 11mpd 146 . . . . . 6 |- (([R ==> [S ==> T]], [R ==> S]), R) |= S
1310simpld 35 . . . . . . 7 |- (([R ==> [S ==> T]], [R ==> S]), R) |= [R ==> [S ==> T]]
145, 9, 13mpd 146 . . . . . 6 |- (([R ==> [S ==> T]], [R ==> S]), R) |= [S ==> T]
151, 12, 14mpd 146 . . . . 5 |- (([R ==> [S ==> T]], [R ==> S]), R) |= T
1615ex 148 . . . 4 |- ([R ==> [S ==> T]], [R ==> S]) |= [R ==> T]
1716ex 148 . . 3 |- [R ==> [S ==> T]] |= [[R ==> S] ==> [R ==> T]]
18 wtru 40 . . 3 |- T.:*
1917, 18adantl 51 . 2 |- (T., [R ==> [S ==> T]]) |= [[R ==> S] ==> [R ==> T]]
2019ex 148 1 |- T. |= [[R ==> [S ==> T]] ==> [[R ==> S] ==> [R ==> T]]]
Colors of variables: type var term
Syntax hints:  *hb 3  T.kt 8  [kbr 9  kct 10   |= wffMMJ2 11  wffMMJ2t 12   ==> tim 111
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
This theorem depends on definitions:  df-ov 65  df-an 118  df-im 119
This theorem is referenced by: (None)
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