ax1 - Higher-Order Logic Explorer

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Theorem ax1 190

Description: Axiom Simp. Axiom A1 of [Margaris] p. 49.

**Hypotheses**
Ref	Expression
ax1.1	⊢ R:∗
ax1.2	⊢ S:∗

**Assertion**
Ref	Expression
ax1	⊢ ⊤⊧[R ⇒ [S ⇒ R]]

**Proof of Theorem ax1**
Step	Hyp	Ref	Expression
1		wtru 40	. . . . 5 ⊢ ⊤:∗
2		ax1.1	. . . . 5 ⊢ R:∗
3	1, 2	simpr 23	. . . 4 ⊢ (⊤, R)⊧R
4		ax1.2	. . . 4 ⊢ S:∗
5	3, 4	adantr 50	. . 3 ⊢ ((⊤, R), S)⊧R
6	5	ex 148	. 2 ⊢ (⊤, R)⊧[S ⇒ R]
7	6	ex 148	1 ⊢ ⊤⊧[R ⇒ [S ⇒ R]]

Colors of variables: type var term

Syntax hints: ∗hb 3 ⊤kt 8 [kbr 9 kct 10 ⊧wffMMJ2 11 wffMMJ2t 12 ⇒ tim 111

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-an 118 df-im 119

This theorem is referenced by: (None)