Theorem pm13.196a

Description: Theorem *13.196 in WhiteheadRussell p. 179. The only difference is the position of the substituted variable. (Contributed by Andrew Salmon, 3-Jun-2011)

	Ref	Expression
Assertion	pm13.196a	⊢ ( ¬ 𝜑 ↔ ∀ 𝑦 ( [ 𝑦 / 𝑥 ] 𝜑 → 𝑦 ≠ 𝑥 ) )

Proof

Step	Hyp	Ref	Expression
1		sbelx	⊢ ( ¬ 𝜑 ↔ ∃ 𝑦 ( 𝑦 = 𝑥 ∧ [ 𝑦 / 𝑥 ] ¬ 𝜑 ) )
2		sbalex	⊢ ( ∃ 𝑦 ( 𝑦 = 𝑥 ∧ [ 𝑦 / 𝑥 ] ¬ 𝜑 ) ↔ ∀ 𝑦 ( 𝑦 = 𝑥 → [ 𝑦 / 𝑥 ] ¬ 𝜑 ) )
3		sbn	⊢ ( [ 𝑦 / 𝑥 ] ¬ 𝜑 ↔ ¬ [ 𝑦 / 𝑥 ] 𝜑 )
4	3	imbi2i	⊢ ( ( 𝑦 = 𝑥 → [ 𝑦 / 𝑥 ] ¬ 𝜑 ) ↔ ( 𝑦 = 𝑥 → ¬ [ 𝑦 / 𝑥 ] 𝜑 ) )
5		con2b	⊢ ( ( 𝑦 = 𝑥 → ¬ [ 𝑦 / 𝑥 ] 𝜑 ) ↔ ( [ 𝑦 / 𝑥 ] 𝜑 → ¬ 𝑦 = 𝑥 ) )
6		df-ne	⊢ ( 𝑦 ≠ 𝑥 ↔ ¬ 𝑦 = 𝑥 )
7	6	bicomi	⊢ ( ¬ 𝑦 = 𝑥 ↔ 𝑦 ≠ 𝑥 )
8	7	imbi2i	⊢ ( ( [ 𝑦 / 𝑥 ] 𝜑 → ¬ 𝑦 = 𝑥 ) ↔ ( [ 𝑦 / 𝑥 ] 𝜑 → 𝑦 ≠ 𝑥 ) )
9	4 5 8	3bitri	⊢ ( ( 𝑦 = 𝑥 → [ 𝑦 / 𝑥 ] ¬ 𝜑 ) ↔ ( [ 𝑦 / 𝑥 ] 𝜑 → 𝑦 ≠ 𝑥 ) )
10	9	albii	⊢ ( ∀ 𝑦 ( 𝑦 = 𝑥 → [ 𝑦 / 𝑥 ] ¬ 𝜑 ) ↔ ∀ 𝑦 ( [ 𝑦 / 𝑥 ] 𝜑 → 𝑦 ≠ 𝑥 ) )
11	1 2 10	3bitri	⊢ ( ¬ 𝜑 ↔ ∀ 𝑦 ( [ 𝑦 / 𝑥 ] 𝜑 → 𝑦 ≠ 𝑥 ) )