Theorem dfov2 75

Description: Reverse 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
dfov2.4	⊢ R⊧((FA)B)

Assertion

Ref	Expression
dfov2	⊢ R⊧[AFB]

Proof of Theorem dfov2

Step	Hyp	Ref	Expression
1		dfov1.1	. . 3 ⊢ F:(α → (β → ∗))
2		dfov1.2	. . 3 ⊢ A:α
3		dfov1.3	. . 3 ⊢ B:β
4	1, 2, 3	wov 72	. 2 ⊢ [AFB]:∗
5		dfov2.4	. 2 ⊢ R⊧((FA)B)
6	5	ax-cb1 29	. . 3 ⊢ R:∗
7	1, 2, 3	df-ov 73	. . 3 ⊢ ⊤⊧(( = [AFB])((FA)B))
8	6, 7	a1i 28	. 2 ⊢ R⊧(( = [AFB])((FA)B))
9	4, 5, 8	mpbirx 53	1 ⊢ R⊧[AFB]

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-jca 17  ax-trud 26  ax-cb1 29  ax-cb2 30  ax-weq 40  ax-refl 42  ax-eqmp 45  ax-wc 49  ax-ceq 51  ax-wov 71

This theorem depends on definitions:  df-ov 73

This theorem is referenced by:  eqcomi  79  eqid  83  ded  84  ceq12  88  leq  91  beta  92  distrc  93  distrl  94  eqtri  95  oveq123  98  hbov  111  ovl  117