Theorem bnj154

Description: Technical lemma for bnj153 . 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	bnj154.1	⊢ ( 𝜑1 ↔ [ 𝑔 / 𝑓 ] 𝜑′ )
	bnj154.2	⊢ ( 𝜑′ ↔ ( 𝑓 ‘ ∅ ) = pred ( 𝑥 , 𝐴 , 𝑅 ) )
Assertion	bnj154	⊢ ( 𝜑1 ↔ ( 𝑔 ‘ ∅ ) = pred ( 𝑥 , 𝐴 , 𝑅 ) )

Proof

Step	Hyp	Ref	Expression
1		bnj154.1	⊢ ( 𝜑1 ↔ [ 𝑔 / 𝑓 ] 𝜑′ )
2		bnj154.2	⊢ ( 𝜑′ ↔ ( 𝑓 ‘ ∅ ) = pred ( 𝑥 , 𝐴 , 𝑅 ) )
3	2	sbcbii	⊢ ( [ 𝑔 / 𝑓 ] 𝜑′ ↔ [ 𝑔 / 𝑓 ] ( 𝑓 ‘ ∅ ) = pred ( 𝑥 , 𝐴 , 𝑅 ) )
4		vex	⊢ 𝑔 ∈ V
5		fveq1	⊢ ( 𝑓 = 𝑔 → ( 𝑓 ‘ ∅ ) = ( 𝑔 ‘ ∅ ) )
6	5	eqeq1d	⊢ ( 𝑓 = 𝑔 → ( ( 𝑓 ‘ ∅ ) = pred ( 𝑥 , 𝐴 , 𝑅 ) ↔ ( 𝑔 ‘ ∅ ) = pred ( 𝑥 , 𝐴 , 𝑅 ) ) )
7	4 6	sbcie	⊢ ( [ 𝑔 / 𝑓 ] ( 𝑓 ‘ ∅ ) = pred ( 𝑥 , 𝐴 , 𝑅 ) ↔ ( 𝑔 ‘ ∅ ) = pred ( 𝑥 , 𝐴 , 𝑅 ) )
8	1 3 7	3bitri	⊢ ( 𝜑1 ↔ ( 𝑔 ‘ ∅ ) = pred ( 𝑥 , 𝐴 , 𝑅 ) )