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

Step	Hyp	Ref	Expression
1		weq 41	. 2 |- = :(* -> (* -> *))
2		ded.2	. . 3 |- (R, T) |= S
3	2	ax-cb2 30	. 2 |- S:*
4		ded.1	. . 3 |- (R, S) |= T
5	4	ax-cb2 30	. 2 |- T:*
6	4, 2	ax-ded 46	. 2 |- R |= (( = S)T)
7	1, 3, 5, 6	dfov2 75	1 |- R |= [S = T]

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