Theorem bnj118

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)

	Ref	Expression
Hypotheses	bnj118.1	⊢ ( 𝜑 ↔ ( 𝑓 ‘ ∅ ) = pred ( 𝑥 , 𝐴 , 𝑅 ) )
	bnj118.2	⊢ ( 𝜑′ ↔ [ 1o / 𝑛 ] 𝜑 )
Assertion	bnj118	⊢ ( 𝜑′ ↔ ( 𝑓 ‘ ∅ ) = pred ( 𝑥 , 𝐴 , 𝑅 ) )

Proof

Step	Hyp	Ref	Expression
1		bnj118.1	⊢ ( 𝜑 ↔ ( 𝑓 ‘ ∅ ) = pred ( 𝑥 , 𝐴 , 𝑅 ) )
2		bnj118.2	⊢ ( 𝜑′ ↔ [ 1o / 𝑛 ] 𝜑 )
3		bnj105	⊢ 1o ∈ V
4	1 3	bnj91	⊢ ( [ 1o / 𝑛 ] 𝜑 ↔ ( 𝑓 ‘ ∅ ) = pred ( 𝑥 , 𝐴 , 𝑅 ) )
5	2 4	bitri	⊢ ( 𝜑′ ↔ ( 𝑓 ‘ ∅ ) = pred ( 𝑥 , 𝐴 , 𝑅 ) )