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Mirrors > Home > HOLE Home > Th. List > ded | GIF version |
Description: Deduction theorem for equality. |
Ref | Expression |
---|---|
ded.1 | ⊢ (R, S)⊧T |
ded.2 | ⊢ (R, T)⊧S |
Ref | Expression |
---|---|
ded | ⊢ R⊧[S = T] |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | weq 38 | . 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 43 | . 2 ⊢ R⊧(( = S)T) |
7 | 1, 3, 5, 6 | dfov2 67 | 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-refl 39 ax-eqmp 42 ax-ded 43 ax-ceq 46 |
This theorem depends on definitions: df-ov 65 |
This theorem is referenced by: dedi 75 eqtru 76 ex 148 notval2 149 dfex2 185 |
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