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Database ZF (ZERMELO-FRAENKEL) SET THEORY ZF Set Theory - start with the Axiom of Extensionality The universal class rspcedeq2vd
Description: Restricted existential specialization, using implicit substitution.
Variant of rspcedvd for equations, in which the right hand side
depends on the quantified variable. (Contributed by AV , 24-Dec-2019)
Ref
Expression
Hypotheses
rspcedeqvd.1 ⊢ ( 𝜑 → 𝐴 ∈ 𝐵 )
rspcedeqvd.2 ⊢ ( ( 𝜑 ∧ 𝑥 = 𝐴 ) → 𝐶 = 𝐷 )
Assertion
rspcedeq2vd ⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝐵 𝐶 = 𝐷 )
Proof
Step
Hyp
Ref
Expression
1
rspcedeqvd.1 ⊢ ( 𝜑 → 𝐴 ∈ 𝐵 )
2
rspcedeqvd.2 ⊢ ( ( 𝜑 ∧ 𝑥 = 𝐴 ) → 𝐶 = 𝐷 )
3
2
eqcomd ⊢ ( ( 𝜑 ∧ 𝑥 = 𝐴 ) → 𝐷 = 𝐶 )
4
3
eqeq2d ⊢ ( ( 𝜑 ∧ 𝑥 = 𝐴 ) → ( 𝐶 = 𝐷 ↔ 𝐶 = 𝐶 ) )
5
eqidd ⊢ ( 𝜑 → 𝐶 = 𝐶 )
6
1 4 5
rspcedvd ⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝐵 𝐶 = 𝐷 )