Theorem wl-sbal2

Description: Move quantifier in and out of substitution. Revised to remove a distinct variable constraint. (Contributed by NM, 2-Jan-2002) Proof is based on wl-sbalnae now. See also sbal2 . (Revised by Wolf Lammen, 25-Jul-2019) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	wl-sbal2	⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → ( [ 𝑧 / 𝑦 ] ∀ 𝑥 𝜑 ↔ ∀ 𝑥 [ 𝑧 / 𝑦 ] 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		naev	⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → ¬ ∀ 𝑥 𝑥 = 𝑧 )
2		wl-sbalnae	⊢ ( ( ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑧 ) → ( [ 𝑧 / 𝑦 ] ∀ 𝑥 𝜑 ↔ ∀ 𝑥 [ 𝑧 / 𝑦 ] 𝜑 ) )
3	1 2	mpdan	⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → ( [ 𝑧 / 𝑦 ] ∀ 𝑥 𝜑 ↔ ∀ 𝑥 [ 𝑧 / 𝑦 ] 𝜑 ) )