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Metamath Proof Explorer

Theorem pm11.53

Description: Theorem *11.53 in WhiteheadRussell p. 164. See pm11.53v for a version requiring fewer axioms. (Contributed by Andrew Salmon, 24-May-2011)

		Ref	Expression
	Assertion	pm11.53	⊢ ( ∀ 𝑥 ∀ 𝑦 ( 𝜑 → 𝜓 ) ↔ ( ∃ 𝑥 𝜑 → ∀ 𝑦 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		19.21v	⊢ ( ∀ 𝑦 ( 𝜑 → 𝜓 ) ↔ ( 𝜑 → ∀ 𝑦 𝜓 ) )
2	1	albii	⊢ ( ∀ 𝑥 ∀ 𝑦 ( 𝜑 → 𝜓 ) ↔ ∀ 𝑥 ( 𝜑 → ∀ 𝑦 𝜓 ) )
3		nfv	⊢ Ⅎ 𝑥 𝜓
4	3	nfal	⊢ Ⅎ 𝑥 ∀ 𝑦 𝜓
5	4	19.23	⊢ ( ∀ 𝑥 ( 𝜑 → ∀ 𝑦 𝜓 ) ↔ ( ∃ 𝑥 𝜑 → ∀ 𝑦 𝜓 ) )
6	2 5	bitri	⊢ ( ∀ 𝑥 ∀ 𝑦 ( 𝜑 → 𝜓 ) ↔ ( ∃ 𝑥 𝜑 → ∀ 𝑦 𝜓 ) )