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SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
Mathbox for BJ
Set theory
Removing some axiom requirements and disjoint variable conditions
bj-rabeqbid
Metamath Proof Explorer
Description: Version of rabeqbidv with two disjoint variable conditions removed and
the third replaced by a nonfreeness hypothesis. (Contributed by BJ , 27-Apr-2019)
Ref
Expression
Hypotheses
bj-rabeqbid.nf
⊢ Ⅎ 𝑥 𝜑
bj-rabeqbid.1
⊢ ( 𝜑 → 𝐴 = 𝐵 )
bj-rabeqbid.2
⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
Assertion
bj-rabeqbid
⊢ ( 𝜑 → { 𝑥 ∈ 𝐴 ∣ 𝜓 } = { 𝑥 ∈ 𝐵 ∣ 𝜒 } )
Proof
Step
Hyp
Ref
Expression
1
bj-rabeqbid.nf
⊢ Ⅎ 𝑥 𝜑
2
bj-rabeqbid.1
⊢ ( 𝜑 → 𝐴 = 𝐵 )
3
bj-rabeqbid.2
⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
4
1 2
bj-rabeqd
⊢ ( 𝜑 → { 𝑥 ∈ 𝐴 ∣ 𝜓 } = { 𝑥 ∈ 𝐵 ∣ 𝜓 } )
5
1 3
rabbid
⊢ ( 𝜑 → { 𝑥 ∈ 𝐵 ∣ 𝜓 } = { 𝑥 ∈ 𝐵 ∣ 𝜒 } )
6
4 5
eqtrd
⊢ ( 𝜑 → { 𝑥 ∈ 𝐴 ∣ 𝜓 } = { 𝑥 ∈ 𝐵 ∣ 𝜒 } )