ax4 - Higher-Order Logic Explorer

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Theorem ax4 140

Description: If A is true for all x:α, then it is true for A.

**Hypothesis**
Ref	Expression
ax4.1	⊢ A:∗

**Assertion**
Ref	Expression
ax4	⊢ (∀λx:α A)⊧A

**Proof of Theorem ax4**
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