Metamath Proof Explorer


Theorem rabeqbidv

Description: Equality of restricted class abstractions. (Contributed by Jeff Madsen, 1-Dec-2009)

Ref Expression
Hypotheses rabeqbidv.1 ( 𝜑𝐴 = 𝐵 )
rabeqbidv.2 ( 𝜑 → ( 𝜓𝜒 ) )
Assertion rabeqbidv ( 𝜑 → { 𝑥𝐴𝜓 } = { 𝑥𝐵𝜒 } )

Proof

Step Hyp Ref Expression
1 rabeqbidv.1 ( 𝜑𝐴 = 𝐵 )
2 rabeqbidv.2 ( 𝜑 → ( 𝜓𝜒 ) )
3 2 adantr ( ( 𝜑𝑥𝐴 ) → ( 𝜓𝜒 ) )
4 1 3 rabeqbidva ( 𝜑 → { 𝑥𝐴𝜓 } = { 𝑥𝐵𝜒 } )