Theorem sbceqal

Description: Class version of one implication of equvelv . (Contributed by Andrew Salmon, 28-Jun-2011)

	Ref	Expression
Assertion	sbceqal	⊢ ( 𝐴 ∈ 𝑉 → ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝑥 = 𝐵 ) → 𝐴 = 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		spsbc	⊢ ( 𝐴 ∈ 𝑉 → ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝑥 = 𝐵 ) → [ 𝐴 / 𝑥 ] ( 𝑥 = 𝐴 → 𝑥 = 𝐵 ) ) )
2		sbcimg	⊢ ( 𝐴 ∈ 𝑉 → ( [ 𝐴 / 𝑥 ] ( 𝑥 = 𝐴 → 𝑥 = 𝐵 ) ↔ ( [ 𝐴 / 𝑥 ] 𝑥 = 𝐴 → [ 𝐴 / 𝑥 ] 𝑥 = 𝐵 ) ) )
3		eqid	⊢ 𝐴 = 𝐴
4		eqsbc3	⊢ ( 𝐴 ∈ 𝑉 → ( [ 𝐴 / 𝑥 ] 𝑥 = 𝐴 ↔ 𝐴 = 𝐴 ) )
5	3 4	mpbiri	⊢ ( 𝐴 ∈ 𝑉 → [ 𝐴 / 𝑥 ] 𝑥 = 𝐴 )
6		pm5.5	⊢ ( [ 𝐴 / 𝑥 ] 𝑥 = 𝐴 → ( ( [ 𝐴 / 𝑥 ] 𝑥 = 𝐴 → [ 𝐴 / 𝑥 ] 𝑥 = 𝐵 ) ↔ [ 𝐴 / 𝑥 ] 𝑥 = 𝐵 ) )
7	5 6	syl	⊢ ( 𝐴 ∈ 𝑉 → ( ( [ 𝐴 / 𝑥 ] 𝑥 = 𝐴 → [ 𝐴 / 𝑥 ] 𝑥 = 𝐵 ) ↔ [ 𝐴 / 𝑥 ] 𝑥 = 𝐵 ) )
8		eqsbc3	⊢ ( 𝐴 ∈ 𝑉 → ( [ 𝐴 / 𝑥 ] 𝑥 = 𝐵 ↔ 𝐴 = 𝐵 ) )
9	2 7 8	3bitrd	⊢ ( 𝐴 ∈ 𝑉 → ( [ 𝐴 / 𝑥 ] ( 𝑥 = 𝐴 → 𝑥 = 𝐵 ) ↔ 𝐴 = 𝐵 ) )
10	1 9	sylibd	⊢ ( 𝐴 ∈ 𝑉 → ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝑥 = 𝐵 ) → 𝐴 = 𝐵 ) )