Metamath Proof Explorer


Theorem raleqbidva

Description: Equality deduction for restricted universal quantifier. (Contributed by Mario Carneiro, 5-Jan-2017)

Ref Expression
Hypotheses raleqbidva.1 φ A = B
raleqbidva.2 φ x A ψ χ
Assertion raleqbidva φ x A ψ x B χ

Proof

Step Hyp Ref Expression
1 raleqbidva.1 φ A = B
2 raleqbidva.2 φ x A ψ χ
3 2 ralbidva φ x A ψ x A χ
4 1 raleqdv φ x A χ x B χ
5 3 4 bitrd φ x A ψ x B χ