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