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Metamath Proof Explorer

Theorem cbvalvw

Description: Change bound variable. Uses only Tarski's FOL axiom schemes. See cbvalv for a version with fewer disjoint variable conditions but requiring more axioms. (Contributed by NM, 9-Apr-2017) (Proof shortened by Wolf Lammen, 28-Feb-2018)

		Ref	Expression
	Hypothesis	cbvalvw.1	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
	Assertion	cbvalvw	⊢ ( ∀ 𝑥 𝜑 ↔ ∀ 𝑦 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		cbvalvw.1	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
2		ax-5	⊢ ( ∀ 𝑥 𝜑 → ∀ 𝑦 ∀ 𝑥 𝜑 )
3		ax-5	⊢ ( ¬ 𝜓 → ∀ 𝑥 ¬ 𝜓 )
4		ax-5	⊢ ( ∀ 𝑦 𝜓 → ∀ 𝑥 ∀ 𝑦 𝜓 )
5		ax-5	⊢ ( ¬ 𝜑 → ∀ 𝑦 ¬ 𝜑 )
6	2 3 4 5 1	cbvalw	⊢ ( ∀ 𝑥 𝜑 ↔ ∀ 𝑦 𝜓 )