Metamath Proof Explorer


Theorem rabeqel

Description: Class element of a restricted class abstraction. (Contributed by Peter Mazsa, 24-Jul-2021)

Ref Expression
Hypotheses rabeqel.1 𝐵 = { 𝑥𝐴𝜑 }
rabeqel.2 ( 𝑥 = 𝐶 → ( 𝜑𝜓 ) )
Assertion rabeqel ( 𝐶𝐵 ↔ ( 𝜓𝐶𝐴 ) )

Proof

Step Hyp Ref Expression
1 rabeqel.1 𝐵 = { 𝑥𝐴𝜑 }
2 rabeqel.2 ( 𝑥 = 𝐶 → ( 𝜑𝜓 ) )
3 2 1 elrab2 ( 𝐶𝐵 ↔ ( 𝐶𝐴𝜓 ) )
4 3 biancomi ( 𝐶𝐵 ↔ ( 𝜓𝐶𝐴 ) )