Theorem bnj609

Description: Technical lemma for bnj852 . 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	bnj609.1	$$⊢\phi \leftrightarrow f\left(\varnothing \right)=pred\left(X,A,R\right)$$
	bnj609.2	No typesetting found for |- ( ph" <-> [. G / f ]. ph ) with typecode |-
	bnj609.3	$$⊢G\in \mathrm{V}$$
Assertion	bnj609	Could not format assertion : No typesetting found for |- ( ph" <-> ( G ` (/) ) = _pred ( X , A , R ) ) with typecode |-

Proof

Step	Hyp	Ref	Expression
1		bnj609.1	$$⊢\phi \leftrightarrow f\left(\varnothing \right)=pred\left(X,A,R\right)$$
2		bnj609.2	Could not format ( ph" <-> [. G / f ]. ph ) : No typesetting found for |- ( ph" <-> [. G / f ]. ph ) with typecode |-
3		bnj609.3	$$⊢G\in \mathrm{V}$$
4		dfsbcq	$$⊢e=G\to \left(\underset{\dot{}}{[}e/f\underset{\dot{}}{]}\phi \leftrightarrow \underset{\dot{}}{[}G/f\underset{\dot{}}{]}\phi \right)$$
5		fveq1	$$⊢e=G\to e\left(\varnothing \right)=G\left(\varnothing \right)$$
6	5	eqeq1d	$$⊢e=G\to \left(e\left(\varnothing \right)=pred\left(X,A,R\right)\leftrightarrow G\left(\varnothing \right)=pred\left(X,A,R\right)\right)$$
7	1	sbcbii	$$⊢\underset{\dot{}}{[}e/f\underset{\dot{}}{]}\phi \leftrightarrow \underset{\dot{}}{[}e/f\underset{\dot{}}{]}f\left(\varnothing \right)=pred\left(X,A,R\right)$$
8		vex	$$⊢e\in \mathrm{V}$$
9		fveq1	$$⊢f=e\to f\left(\varnothing \right)=e\left(\varnothing \right)$$
10	9	eqeq1d	$$⊢f=e\to \left(f\left(\varnothing \right)=pred\left(X,A,R\right)\leftrightarrow e\left(\varnothing \right)=pred\left(X,A,R\right)\right)$$
11	8 10	sbcie	$$⊢\underset{\dot{}}{[}e/f\underset{\dot{}}{]}f\left(\varnothing \right)=pred\left(X,A,R\right)\leftrightarrow e\left(\varnothing \right)=pred\left(X,A,R\right)$$
12	7 11	bitri	$$⊢\underset{\dot{}}{[}e/f\underset{\dot{}}{]}\phi \leftrightarrow e\left(\varnothing \right)=pred\left(X,A,R\right)$$
13	3 4 6 12	vtoclb	$$⊢\underset{\dot{}}{[}G/f\underset{\dot{}}{]}\phi \leftrightarrow G\left(\varnothing \right)=pred\left(X,A,R\right)$$
14	2 13	bitri	Could not format ( ph" <-> ( G ` (/) ) = _pred ( X , A , R ) ) : No typesetting found for |- ( ph" <-> ( G ` (/) ) = _pred ( X , A , R ) ) with typecode |-