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Database ZF (ZERMELO-FRAENKEL) SET THEORY ZF Set Theory - start with the Axiom of Extensionality The universal class spcgv
Description: Rule of specialization, using implicit substitution. Compare Theorem
7.3 of Quine p. 44. (Contributed by NM , 22-Jun-1994) Avoid
ax-10 , ax-11 . (Revised by Wolf Lammen , 25-Aug-2023)
Ref
Expression
Hypothesis
spcgv.1 ⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
Assertion
spcgv ⊢ ( 𝐴 ∈ 𝑉 → ( ∀ 𝑥 𝜑 → 𝜓 ) )
Proof
Step
Hyp
Ref
Expression
1
spcgv.1 ⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
2
elex ⊢ ( 𝐴 ∈ 𝑉 → 𝐴 ∈ V )
3
id ⊢ ( 𝐴 ∈ V → 𝐴 ∈ V )
4
1
adantl ⊢ ( ( 𝐴 ∈ V ∧ 𝑥 = 𝐴 ) → ( 𝜑 ↔ 𝜓 ) )
5
3 4
spcdv ⊢ ( 𝐴 ∈ V → ( ∀ 𝑥 𝜑 → 𝜓 ) )
6
2 5
syl ⊢ ( 𝐴 ∈ 𝑉 → ( ∀ 𝑥 𝜑 → 𝜓 ) )