Theorem bnj133

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

	Ref	Expression
Hypotheses	bnj133.1	⊢ ( 𝜑 ↔ ∃ 𝑥 𝜓 )
	bnj133.2	⊢ ( 𝜒 ↔ 𝜓 )
Assertion	bnj133	⊢ ( 𝜑 ↔ ∃ 𝑥 𝜒 )

Proof

Step	Hyp	Ref	Expression
1		bnj133.1	⊢ ( 𝜑 ↔ ∃ 𝑥 𝜓 )
2		bnj133.2	⊢ ( 𝜒 ↔ 𝜓 )
3	2	exbii	⊢ ( ∃ 𝑥 𝜒 ↔ ∃ 𝑥 𝜓 )
4	1 3	bitr4i	⊢ ( 𝜑 ↔ ∃ 𝑥 𝜒 )