Theorem elrabrd

Description: Deduction version of elrab , just like elrabd , but backwards direction. (Contributed by Thierry Arnoux, 15-Jan-2026)

	Ref	Expression
Hypotheses	elrabrd.1	⊢ ( 𝑥 = 𝐴 → ( 𝜓 ↔ 𝜒 ) )
	elrabrd.2	⊢ ( 𝜑 → 𝐴 ∈ { 𝑥 ∈ 𝐵 ∣ 𝜓 } )
Assertion	elrabrd	⊢ ( 𝜑 → 𝜒 )

Proof

Step	Hyp	Ref	Expression
1		elrabrd.1	⊢ ( 𝑥 = 𝐴 → ( 𝜓 ↔ 𝜒 ) )
2		elrabrd.2	⊢ ( 𝜑 → 𝐴 ∈ { 𝑥 ∈ 𝐵 ∣ 𝜓 } )
3	1	elrab	⊢ ( 𝐴 ∈ { 𝑥 ∈ 𝐵 ∣ 𝜓 } ↔ ( 𝐴 ∈ 𝐵 ∧ 𝜒 ) )
4	2 3	sylib	⊢ ( 𝜑 → ( 𝐴 ∈ 𝐵 ∧ 𝜒 ) )
5	4	simprd	⊢ ( 𝜑 → 𝜒 )