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Mirrors > Home > HOLE Home > Th. List > ax4 | GIF version |
Description: If A is true for all x:α, then it is true for A. |
Ref | Expression |
---|---|
ax4.1 | ⊢ A:∗ |
Ref | Expression |
---|---|
ax4 | ⊢ (∀λx:α A)⊧A |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax4.1 | . . . 4 ⊢ A:∗ | |
2 | 1 | wl 59 | . . 3 ⊢ λx:α A:(α → ∗) |
3 | wv 58 | . . 3 ⊢ x:α:α | |
4 | 2, 3 | ax4g 139 | . 2 ⊢ (∀λx:α A)⊧(λx:α Ax:α) |
5 | 4 | ax-cb1 29 | . . 3 ⊢ (∀λx:α A):∗ |
6 | 1 | beta 82 | . . 3 ⊢ ⊤⊧[(λx:α Ax:α) = A] |
7 | 5, 6 | a1i 28 | . 2 ⊢ (∀λx:α A)⊧[(λx:α Ax:α) = A] |
8 | 4, 7 | mpbi 72 | 1 ⊢ (∀λx:α A)⊧A |
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: alimdv 172 alnex 174 ax5 194 ax7 196 ax10 200 axext 206 |
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