19.8a - Higher-Order Logic Explorer

	Higher-Order Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > HOLE Home > Th. List > 19.8a		GIF version

Theorem 19.8a 160

Description: Existential introduction.

**Hypothesis**
Ref	Expression
19.8a.1	⊢ A:∗

**Assertion**
Ref	Expression
19.8a	⊢ A⊧(∃λx:α A)

**Proof of Theorem 19.8a**
Step	Hyp	Ref	Expression
1		19.8a.1	. . . 4 ⊢ A:∗
2	1	ax-id 24	. . 3 ⊢ A⊧A
3	1	beta 82	. . . 4 ⊢ ⊤⊧[(λx:α Ax:α) = A]
4	1, 3	a1i 28	. . 3 ⊢ A⊧[(λx:α Ax:α) = A]
5	2, 4	mpbir 77	. 2 ⊢ A⊧(λx:α Ax:α)
6	1	wl 59	. . 3 ⊢ λx:α A:(α → ∗)
7		wv 58	. . 3 ⊢ x:α:α
8	6, 7	ax4e 158	. 2 ⊢ (λx:α Ax:α)⊧(∃λx:α A)
9	5, 8	syl 16	1 ⊢ A⊧(∃λx:α 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 ∃tex 113

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-distrl 63 ax-hbl1 93 ax-17 95 ax-inst 103

This theorem depends on definitions: df-ov 65 df-al 116 df-an 118 df-im 119 df-ex 121

This theorem is referenced by: eximdv 173 alnex 174 ax9 199