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Mirrors > Home > HOLE Home > Th. List > cla4v | GIF version |
Description: If A(x) is true for all x:α, then it is true for C = A(B). |
Ref | Expression |
---|---|
cla4v.1 | ⊢ A:∗ |
cla4v.2 | ⊢ B:α |
cla4v.3 | ⊢ [x:α = B]⊧[A = C] |
Ref | Expression |
---|---|
cla4v | ⊢ (∀λx:α A)⊧C |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cla4v.1 | . . . 4 ⊢ A:∗ | |
2 | 1 | wl 59 | . . 3 ⊢ λx:α A:(α → ∗) |
3 | cla4v.2 | . . 3 ⊢ B:α | |
4 | 2, 3 | ax4g 139 | . 2 ⊢ (∀λx:α A)⊧(λx:α AB) |
5 | 4 | ax-cb1 29 | . . 3 ⊢ (∀λx:α A):∗ |
6 | cla4v.3 | . . . 4 ⊢ [x:α = B]⊧[A = C] | |
7 | 1, 3, 6 | cl 106 | . . 3 ⊢ ⊤⊧[(λx:α AB) = C] |
8 | 5, 7 | a1i 28 | . 2 ⊢ (∀λx:α A)⊧[(λx:α AB) = C] |
9 | 4, 8 | mpbi 72 | 1 ⊢ (∀λx:α A)⊧C |
Colors of variables: type var term |
Syntax hints: tv 1 ∗hb 3 kc 5 λkl 6 = ke 7 [kbr 9 ⊧wffMMJ2 11 wffMMJ2t 12 ∀tal 112 |
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-hbl1 93 ax-17 95 ax-inst 103 |
This theorem depends on definitions: df-ov 65 df-al 116 |
This theorem is referenced by: pm2.21 143 ecase 153 exlimdv2 156 ax4e 158 eta 166 exlimd 171 ac 184 ax10 200 |
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