Theorem bnj934

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	bnj934.1	⊢ ( 𝜑 ↔ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) )
	bnj934.4	⊢ ( 𝜑′ ↔ [ 𝑝 / 𝑛 ] 𝜑 )
	bnj934.7	⊢ ( 𝜑″ ↔ [ 𝐺 / 𝑓 ] 𝜑′ )
	bnj934.50	⊢ 𝐺 ∈ V
Assertion	bnj934	⊢ ( ( 𝜑 ∧ ( 𝐺 ‘ ∅ ) = ( 𝑓 ‘ ∅ ) ) → 𝜑″ )

Proof

Step	Hyp	Ref	Expression
1		bnj934.1	⊢ ( 𝜑 ↔ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) )
2		bnj934.4	⊢ ( 𝜑′ ↔ [ 𝑝 / 𝑛 ] 𝜑 )
3		bnj934.7	⊢ ( 𝜑″ ↔ [ 𝐺 / 𝑓 ] 𝜑′ )
4		bnj934.50	⊢ 𝐺 ∈ V
5		eqtr	⊢ ( ( ( 𝐺 ‘ ∅ ) = ( 𝑓 ‘ ∅ ) ∧ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ) → ( 𝐺 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) )
6	1 5	sylan2b	⊢ ( ( ( 𝐺 ‘ ∅ ) = ( 𝑓 ‘ ∅ ) ∧ 𝜑 ) → ( 𝐺 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) )
7		vex	⊢ 𝑝 ∈ V
8	1 2 7	bnj523	⊢ ( 𝜑′ ↔ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) )
9	8 1	bitr4i	⊢ ( 𝜑′ ↔ 𝜑 )
10	9	sbcbii	⊢ ( [ 𝐺 / 𝑓 ] 𝜑′ ↔ [ 𝐺 / 𝑓 ] 𝜑 )
11	3 10	bitri	⊢ ( 𝜑″ ↔ [ 𝐺 / 𝑓 ] 𝜑 )
12	1 11 4	bnj609	⊢ ( 𝜑″ ↔ ( 𝐺 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) )
13	6 12	sylibr	⊢ ( ( ( 𝐺 ‘ ∅ ) = ( 𝑓 ‘ ∅ ) ∧ 𝜑 ) → 𝜑″ )
14	13	ancoms	⊢ ( ( 𝜑 ∧ ( 𝐺 ‘ ∅ ) = ( 𝑓 ‘ ∅ ) ) → 𝜑″ )