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Metamath Proof Explorer

Theorem bnj538

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.) (Proof shortened by OpenAI, 30-Mar-2020)

		Ref	Expression
	Hypothesis	bnj538.1	⊢ 𝐴 ∈ V
	Assertion	bnj538	⊢ ( [ 𝐴 / 𝑦 ] ∀ 𝑥 ∈ 𝐵 𝜑 ↔ ∀ 𝑥 ∈ 𝐵 [ 𝐴 / 𝑦 ] 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		bnj538.1	⊢ 𝐴 ∈ V
2		sbcralg	⊢ ( 𝐴 ∈ V → ( [ 𝐴 / 𝑦 ] ∀ 𝑥 ∈ 𝐵 𝜑 ↔ ∀ 𝑥 ∈ 𝐵 [ 𝐴 / 𝑦 ] 𝜑 ) )
3	1 2	ax-mp	⊢ ( [ 𝐴 / 𝑦 ] ∀ 𝑥 ∈ 𝐵 𝜑 ↔ ∀ 𝑥 ∈ 𝐵 [ 𝐴 / 𝑦 ] 𝜑 )