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	⊢ ( 𝜑 ↔ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) )
	bnj609.2	⊢ ( 𝜑″ ↔ [ 𝐺 / 𝑓 ] 𝜑 )
	bnj609.3	⊢ 𝐺 ∈ V
Assertion	bnj609	⊢ ( 𝜑″ ↔ ( 𝐺 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) )

Proof

Step	Hyp	Ref	Expression
1		bnj609.1	⊢ ( 𝜑 ↔ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) )
2		bnj609.2	⊢ ( 𝜑″ ↔ [ 𝐺 / 𝑓 ] 𝜑 )
3		bnj609.3	⊢ 𝐺 ∈ V
4		dfsbcq	⊢ ( 𝑒 = 𝐺 → ( [ 𝑒 / 𝑓 ] 𝜑 ↔ [ 𝐺 / 𝑓 ] 𝜑 ) )
5		fveq1	⊢ ( 𝑒 = 𝐺 → ( 𝑒 ‘ ∅ ) = ( 𝐺 ‘ ∅ ) )
6	5	eqeq1d	⊢ ( 𝑒 = 𝐺 → ( ( 𝑒 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ↔ ( 𝐺 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ) )
7	1	sbcbii	⊢ ( [ 𝑒 / 𝑓 ] 𝜑 ↔ [ 𝑒 / 𝑓 ] ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) )
8		vex	⊢ 𝑒 ∈ V
9		fveq1	⊢ ( 𝑓 = 𝑒 → ( 𝑓 ‘ ∅ ) = ( 𝑒 ‘ ∅ ) )
10	9	eqeq1d	⊢ ( 𝑓 = 𝑒 → ( ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ↔ ( 𝑒 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ) )
11	8 10	sbcie	⊢ ( [ 𝑒 / 𝑓 ] ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ↔ ( 𝑒 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) )
12	7 11	bitri	⊢ ( [ 𝑒 / 𝑓 ] 𝜑 ↔ ( 𝑒 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) )
13	3 4 6 12	vtoclb	⊢ ( [ 𝐺 / 𝑓 ] 𝜑 ↔ ( 𝐺 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) )
14	2 13	bitri	⊢ ( 𝜑″ ↔ ( 𝐺 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) )