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Mirrors > Home > HOLE Home > Th. List > alrimiv | GIF version |
Description: If one can prove R⊧A where R does not contain x, then A is true for all x. |
Ref | Expression |
---|---|
alrimiv.1 | ⊢ R⊧A |
Ref | Expression |
---|---|
alrimiv | ⊢ R⊧(∀λx:α A) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | alrimiv.1 | . . . 4 ⊢ R⊧A | |
2 | 1 | ax-cb2 30 | . . 3 ⊢ A:∗ |
3 | wtru 40 | . . . 4 ⊢ ⊤:∗ | |
4 | 1 | eqtru 76 | . . . 4 ⊢ R⊧[⊤ = A] |
5 | 3, 4 | eqcomi 70 | . . 3 ⊢ R⊧[A = ⊤] |
6 | 2, 5 | leq 81 | . 2 ⊢ R⊧[λx:α A = λx:α ⊤] |
7 | 1 | ax-cb1 29 | . . 3 ⊢ R:∗ |
8 | 2 | wl 59 | . . . 4 ⊢ λx:α A:(α → ∗) |
9 | 8 | alval 132 | . . 3 ⊢ ⊤⊧[(∀λx:α A) = [λx:α A = λx:α ⊤]] |
10 | 7, 9 | a1i 28 | . 2 ⊢ R⊧[(∀λx:α A) = [λx:α A = λx:α ⊤]] |
11 | 6, 10 | mpbir 77 | 1 ⊢ R⊧(∀λx:α A) |
Colors of variables: type var term |
Syntax hints: ∗hb 3 kc 5 λkl 6 = ke 7 ⊤kt 8 [kbr 9 ⊧wffMMJ2 11 ∀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: exlimdv2 156 ax4e 158 exlimd 171 axgen 197 ax10 200 ax11 201 axrep 207 axpow 208 axun 209 |
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