Theorem pm10.542

Description: Theorem *10.542 in WhiteheadRussell p. 156. (Contributed by Andrew Salmon, 24-May-2011)

	Ref	Expression
Assertion	pm10.542	⊢ ( ∀ 𝑥 ( 𝜑 → ( 𝜒 → 𝜓 ) ) ↔ ( 𝜒 → ∀ 𝑥 ( 𝜑 → 𝜓 ) ) )

Proof

Step	Hyp	Ref	Expression
1		bi2.04	⊢ ( ( 𝜑 → ( 𝜒 → 𝜓 ) ) ↔ ( 𝜒 → ( 𝜑 → 𝜓 ) ) )
2	1	albii	⊢ ( ∀ 𝑥 ( 𝜑 → ( 𝜒 → 𝜓 ) ) ↔ ∀ 𝑥 ( 𝜒 → ( 𝜑 → 𝜓 ) ) )
3		19.21v	⊢ ( ∀ 𝑥 ( 𝜒 → ( 𝜑 → 𝜓 ) ) ↔ ( 𝜒 → ∀ 𝑥 ( 𝜑 → 𝜓 ) ) )
4	2 3	bitri	⊢ ( ∀ 𝑥 ( 𝜑 → ( 𝜒 → 𝜓 ) ) ↔ ( 𝜒 → ∀ 𝑥 ( 𝜑 → 𝜓 ) ) )