Theorem dfov1 74

Description: Forward direction of df-ov 73. (Contributed by Mario Carneiro, 8-Oct-2014.)

Hypotheses

Ref	Expression
dfov1.1	⊢ F:(α → (β → ∗))
dfov1.2	⊢ A:α
dfov1.3	⊢ B:β
dfov1.4	⊢ R⊧[AFB]

Assertion

Ref	Expression
dfov1	⊢ R⊧((FA)B)

Proof of Theorem dfov1

Step	Hyp	Ref	Expression
1		dfov1.4	. 2 ⊢ R⊧[AFB]
2	1	ax-cb1 29	. . 3 ⊢ R:∗
3		dfov1.1	. . . 4 ⊢ F:(α → (β → ∗))
4		dfov1.2	. . . 4 ⊢ A:α
5		dfov1.3	. . . 4 ⊢ B:β
6	3, 4, 5	df-ov 73	. . 3 ⊢ ⊤⊧(( = [AFB])((FA)B))
7	2, 6	a1i 28	. 2 ⊢ R⊧(( = [AFB])((FA)B))
8	1, 7	ax-eqmp 45	1 ⊢ R⊧((FA)B)

Syntax hints:   → ht 2  ∗hb 3  kc 5   = ke 7  [kbr 9  ⊧wffMMJ2 11  wffMMJ2t 12

This theorem was proved from axioms:  ax-syl 15  ax-trud 26  ax-cb1 29  ax-eqmp 45

This theorem depends on definitions:  df-ov 73

This theorem is referenced by:  eqcomi  79  mpbi  82  ceq12  88  leq  91  eqtri  95