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Theorem ded 84
Description: Deduction theorem for equality. (Contributed by Mario Carneiro, 7-Oct-2014.)
Hypotheses
Ref Expression
ded.1 |- (R, S) |= T
ded.2 |- (R, T) |= S
Assertion
Ref Expression
ded |- R |= [S = T]

Proof of Theorem ded
StepHypRef Expression
1 weq 41 . 2 |- = :(* -> (* -> *))
2 ded.2 . . 3 |- (R, T) |= S
32ax-cb2 30 . 2 |- S:*
4 ded.1 . . 3 |- (R, S) |= T
54ax-cb2 30 . 2 |- T:*
64, 2ax-ded 46 . 2 |- R |= (( = S)T)
71, 3, 5, 6dfov2 75 1 |- R |= [S = T]
Colors of variables: type var term
Syntax hints:  *hb 3   = ke 7  [kbr 9  kct 10   |= wffMMJ2 11
This theorem was proved from axioms:  ax-syl 15  ax-jca 17  ax-trud 26  ax-cb1 29  ax-cb2 30  ax-weq 40  ax-refl 42  ax-eqmp 45  ax-ded 46  ax-wc 49  ax-ceq 51  ax-wov 71
This theorem depends on definitions:  df-ov 73
This theorem is referenced by:  dedi  85  eqtru  86  ex  158  notval2  159  dfex2  198
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