Theorem pm13.194

Description: Theorem *13.194 in WhiteheadRussell p. 179. (Contributed by Andrew Salmon, 3-Jun-2011)

	Ref	Expression
Assertion	pm13.194	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) ↔ ( [ 𝑦 / 𝑥 ] 𝜑 ∧ 𝜑 ∧ 𝑥 = 𝑦 ) )

Proof

Step	Hyp	Ref	Expression
1		pm13.13a	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → [ 𝑦 / 𝑥 ] 𝜑 )
2		sbsbc	⊢ ( [ 𝑦 / 𝑥 ] 𝜑 ↔ [ 𝑦 / 𝑥 ] 𝜑 )
3	1 2	sylibr	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → [ 𝑦 / 𝑥 ] 𝜑 )
4		simpl	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → 𝜑 )
5		simpr	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → 𝑥 = 𝑦 )
6	3 4 5	3jca	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( [ 𝑦 / 𝑥 ] 𝜑 ∧ 𝜑 ∧ 𝑥 = 𝑦 ) )
7		3simpc	⊢ ( ( [ 𝑦 / 𝑥 ] 𝜑 ∧ 𝜑 ∧ 𝑥 = 𝑦 ) → ( 𝜑 ∧ 𝑥 = 𝑦 ) )
8	6 7	impbii	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) ↔ ( [ 𝑦 / 𝑥 ] 𝜑 ∧ 𝜑 ∧ 𝑥 = 𝑦 ) )