Metamath Proof Explorer


Theorem bnj984

Description: Technical lemma for bnj69 . This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)

Ref Expression
Hypotheses bnj984.3 χ n D f Fn n φ ψ
bnj984.11 B = f | n D f Fn n φ ψ
Assertion bnj984 G A G B [˙G / f]˙ n χ

Proof

Step Hyp Ref Expression
1 bnj984.3 χ n D f Fn n φ ψ
2 bnj984.11 B = f | n D f Fn n φ ψ
3 2 eleq2i G B G f | n D f Fn n φ ψ
4 sbc8g G A [˙G / f]˙ n D f Fn n φ ψ G f | n D f Fn n φ ψ
5 3 4 bitr4id G A G B [˙G / f]˙ n D f Fn n φ ψ
6 df-rex n D f Fn n φ ψ n n D f Fn n φ ψ
7 bnj252 n D f Fn n φ ψ n D f Fn n φ ψ
8 1 7 bitri χ n D f Fn n φ ψ
9 6 8 bnj133 n D f Fn n φ ψ n χ
10 9 sbcbii [˙G / f]˙ n D f Fn n φ ψ [˙G / f]˙ n χ
11 5 10 bitrdi G A G B [˙G / f]˙ n χ