Theorem alrimi 182

Description: If one can prove R⊧A where R does not contain x, then A is true for all x. (Contributed by Mario Carneiro, 9-Oct-2014.)

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 43	. . . 4 ⊢ ⊤:∗
4	1	eqtru 86	. . . 4 ⊢ R⊧[⊤ = A]
5	3, 4	eqcomi 79	. . 3 ⊢ R⊧[A = ⊤]
6		alrimi.2	. . 3 ⊢ ⊤⊧[(λx:α Ry:α) = R]
7	2, 5, 6	leqf 181	. 2 ⊢ R⊧[λx:α A = λx:α ⊤]
8	1	ax-cb1 29	. . 3 ⊢ R:∗
9	2	wl 66	. . . 4 ⊢ λx:α A:(α → ∗)
10	9	alval 142	. . 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 87	1 ⊢ R⊧(∀λx:α A)

Syntax hints:  tv 1  ∗hb 3  kc 5  λkl 6   = ke 7  ⊤kt 8  [kbr 9  ⊧wffMMJ2 11  ∀tal 122

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-wctl 31  ax-wctr 32  ax-weq 40  ax-refl 42  ax-eqmp 45  ax-ded 46  ax-wct 47  ax-wc 49  ax-ceq 51  ax-wv 63  ax-wl 65  ax-beta 67  ax-distrc 68  ax-leq 69  ax-wov 71  ax-eqtypi 77  ax-eqtypri 80  ax-hbl1 103  ax-17 105  ax-inst 113  ax-eta 177

This theorem depends on definitions:  df-ov 73  df-al 126

This theorem is referenced by:  alimdv  184  alnex  186  isfree  188  ax5  207  ax7  209