Theorem alrimiv 151

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.)

Hypothesis

Ref	Expression
alrimiv.1	⊢ R⊧A

Assertion

Ref	Expression
alrimiv	⊢ R⊧(∀λx:α A)

Distinct variable groups:   x,R   α,x

Proof of Theorem alrimiv

Step	Hyp	Ref	Expression
1		alrimiv.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	2, 5	leq 91	. 2 ⊢ R⊧[λx:α A = λx:α ⊤]
7	1	ax-cb1 29	. . 3 ⊢ R:∗
8	2	wl 66	. . . 4 ⊢ λx:α A:(α → ∗)
9	8	alval 142	. . 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 87	1 ⊢ R⊧(∀λx:α A)

Syntax hints:  ∗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

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

This theorem is referenced by:  exlimdv2  166  ax4e  168  exlimd  183  axgen  210  ax10  213  ax11  214  axrep  220  axpow  221  axun  222