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Metamath Proof Explorer

Theorem cbvral4vw

Description: Change bound variables of quadruple restricted universal quantification, using implicit substitution. (Contributed by Scott Fenton, 2-Mar-2025)

		Ref	Expression
	Hypotheses	cbvral4vw.1	⊢ ( 𝑥 = 𝑎 → ( 𝜑 ↔ 𝜒 ) )
		cbvral4vw.2	⊢ ( 𝑦 = 𝑏 → ( 𝜒 ↔ 𝜃 ) )
		cbvral4vw.3	⊢ ( 𝑧 = 𝑐 → ( 𝜃 ↔ 𝜏 ) )
		cbvral4vw.4	⊢ ( 𝑤 = 𝑑 → ( 𝜏 ↔ 𝜓 ) )
	Assertion	cbvral4vw	⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐶 ∀ 𝑤 ∈ 𝐷 𝜑 ↔ ∀ 𝑎 ∈ 𝐴 ∀ 𝑏 ∈ 𝐵 ∀ 𝑐 ∈ 𝐶 ∀ 𝑑 ∈ 𝐷 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		cbvral4vw.1	⊢ ( 𝑥 = 𝑎 → ( 𝜑 ↔ 𝜒 ) )
2		cbvral4vw.2	⊢ ( 𝑦 = 𝑏 → ( 𝜒 ↔ 𝜃 ) )
3		cbvral4vw.3	⊢ ( 𝑧 = 𝑐 → ( 𝜃 ↔ 𝜏 ) )
4		cbvral4vw.4	⊢ ( 𝑤 = 𝑑 → ( 𝜏 ↔ 𝜓 ) )
5	1	ralbidv	⊢ ( 𝑥 = 𝑎 → ( ∀ 𝑤 ∈ 𝐷 𝜑 ↔ ∀ 𝑤 ∈ 𝐷 𝜒 ) )
6	2	ralbidv	⊢ ( 𝑦 = 𝑏 → ( ∀ 𝑤 ∈ 𝐷 𝜒 ↔ ∀ 𝑤 ∈ 𝐷 𝜃 ) )
7	3	ralbidv	⊢ ( 𝑧 = 𝑐 → ( ∀ 𝑤 ∈ 𝐷 𝜃 ↔ ∀ 𝑤 ∈ 𝐷 𝜏 ) )
8	5 6 7	cbvral3vw	⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐶 ∀ 𝑤 ∈ 𝐷 𝜑 ↔ ∀ 𝑎 ∈ 𝐴 ∀ 𝑏 ∈ 𝐵 ∀ 𝑐 ∈ 𝐶 ∀ 𝑤 ∈ 𝐷 𝜏 )
9	4	cbvralvw	⊢ ( ∀ 𝑤 ∈ 𝐷 𝜏 ↔ ∀ 𝑑 ∈ 𝐷 𝜓 )
10	9	3ralbii	⊢ ( ∀ 𝑎 ∈ 𝐴 ∀ 𝑏 ∈ 𝐵 ∀ 𝑐 ∈ 𝐶 ∀ 𝑤 ∈ 𝐷 𝜏 ↔ ∀ 𝑎 ∈ 𝐴 ∀ 𝑏 ∈ 𝐵 ∀ 𝑐 ∈ 𝐶 ∀ 𝑑 ∈ 𝐷 𝜓 )
11	8 10	bitri	⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐶 ∀ 𝑤 ∈ 𝐷 𝜑 ↔ ∀ 𝑎 ∈ 𝐴 ∀ 𝑏 ∈ 𝐵 ∀ 𝑐 ∈ 𝐶 ∀ 𝑑 ∈ 𝐷 𝜓 )