Theorem alrimi 170

Description: If one can prove R⊧A where R does not contain x, then A is true for all x.

Hypotheses

Ref	Expression
alrimi.1	⊢ R⊧A
alrimi.2	⊢ ⊤⊧[(λx:α Ry:α) = R]

Assertion

Ref	Expression
alrimi	⊢ R⊧(∀λx:α A)

Distinct variable groups:   y,A   y,R   x,y,α

Proof of Theorem alrimi

Step	Hyp	Ref	Expression
1		alrimi.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		alrimi.2	. . 3 ⊢ ⊤⊧[(λx:α Ry:α) = R]
7	2, 5, 6	leqf 169	. 2 ⊢ R⊧[λx:α A = λx:α ⊤]
8	1	ax-cb1 29	. . 3 ⊢ R:∗
9	2	wl 59	. . . 4 ⊢ λx:α A:(α → ∗)
10	9	alval 132	. . 3 ⊢ ⊤⊧[(∀λx:α A) = [λx:α A = λx:α ⊤]]
11	8, 10	a1i 28	. 2 ⊢ R⊧[(∀λx:α A) = [λx:α A = λx:α ⊤]]
12	7, 11	mpbir 77	1 ⊢ R⊧(∀λx:α A)

Syntax hints:  tv 1  ∗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  ax-eta 165

This theorem depends on definitions:  df-ov 65  df-al 116

This theorem is referenced by:  alimdv  172  alnex  174  isfree  176  ax5  194  ax7  196