Theorem bnj125

Description: Technical lemma for bnj150 . 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	bnj125.1	⊢ ( 𝜑 ↔ ( 𝑓 ‘ ∅ ) = pred ( 𝑥 , 𝐴 , 𝑅 ) )
	bnj125.2	⊢ ( 𝜑′ ↔ [ 1o / 𝑛 ] 𝜑 )
	bnj125.3	⊢ ( 𝜑″ ↔ [ 𝐹 / 𝑓 ] 𝜑′ )
	bnj125.4	⊢ 𝐹 = { ⟨ ∅ , pred ( 𝑥 , 𝐴 , 𝑅 ) ⟩ }
Assertion	bnj125	⊢ ( 𝜑″ ↔ ( 𝐹 ‘ ∅ ) = pred ( 𝑥 , 𝐴 , 𝑅 ) )

Proof

Step	Hyp	Ref	Expression
1		bnj125.1	⊢ ( 𝜑 ↔ ( 𝑓 ‘ ∅ ) = pred ( 𝑥 , 𝐴 , 𝑅 ) )
2		bnj125.2	⊢ ( 𝜑′ ↔ [ 1o / 𝑛 ] 𝜑 )
3		bnj125.3	⊢ ( 𝜑″ ↔ [ 𝐹 / 𝑓 ] 𝜑′ )
4		bnj125.4	⊢ 𝐹 = { ⟨ ∅ , pred ( 𝑥 , 𝐴 , 𝑅 ) ⟩ }
5	2	sbcbii	⊢ ( [ 𝐹 / 𝑓 ] 𝜑′ ↔ [ 𝐹 / 𝑓 ] [ 1o / 𝑛 ] 𝜑 )
6		bnj105	⊢ 1o ∈ V
7	1 6	bnj91	⊢ ( [ 1o / 𝑛 ] 𝜑 ↔ ( 𝑓 ‘ ∅ ) = pred ( 𝑥 , 𝐴 , 𝑅 ) )
8	7	sbcbii	⊢ ( [ 𝐹 / 𝑓 ] [ 1o / 𝑛 ] 𝜑 ↔ [ 𝐹 / 𝑓 ] ( 𝑓 ‘ ∅ ) = pred ( 𝑥 , 𝐴 , 𝑅 ) )
9	4	bnj95	⊢ 𝐹 ∈ V
10		fveq1	⊢ ( 𝑓 = 𝐹 → ( 𝑓 ‘ ∅ ) = ( 𝐹 ‘ ∅ ) )
11	10	eqeq1d	⊢ ( 𝑓 = 𝐹 → ( ( 𝑓 ‘ ∅ ) = pred ( 𝑥 , 𝐴 , 𝑅 ) ↔ ( 𝐹 ‘ ∅ ) = pred ( 𝑥 , 𝐴 , 𝑅 ) ) )
12	9 11	sbcie	⊢ ( [ 𝐹 / 𝑓 ] ( 𝑓 ‘ ∅ ) = pred ( 𝑥 , 𝐴 , 𝑅 ) ↔ ( 𝐹 ‘ ∅ ) = pred ( 𝑥 , 𝐴 , 𝑅 ) )
13	8 12	bitri	⊢ ( [ 𝐹 / 𝑓 ] [ 1o / 𝑛 ] 𝜑 ↔ ( 𝐹 ‘ ∅ ) = pred ( 𝑥 , 𝐴 , 𝑅 ) )
14	5 13	bitri	⊢ ( [ 𝐹 / 𝑓 ] 𝜑′ ↔ ( 𝐹 ‘ ∅ ) = pred ( 𝑥 , 𝐴 , 𝑅 ) )
15	3 14	bitri	⊢ ( 𝜑″ ↔ ( 𝐹 ‘ ∅ ) = pred ( 𝑥 , 𝐴 , 𝑅 ) )