Theorem bj-vjust

Description: Justification theorem for bj-df-v . See also vjust . (Contributed by BJ, 30-Nov-2019) (Proof modification is discouraged.)

	Ref	Expression
Assertion	bj-vjust	⊢ { 𝑥 ∣ ⊤ } = { 𝑦 ∣ ⊤ }

Proof

Step	Hyp	Ref	Expression
1		df-clab	⊢ ( 𝑧 ∈ { 𝑥 ∣ ⊤ } ↔ [ 𝑧 / 𝑥 ] ⊤ )
2		sbv	⊢ ( [ 𝑧 / 𝑦 ] ⊤ ↔ ⊤ )
3		df-clab	⊢ ( 𝑧 ∈ { 𝑦 ∣ ⊤ } ↔ [ 𝑧 / 𝑦 ] ⊤ )
4		sbv	⊢ ( [ 𝑧 / 𝑥 ] ⊤ ↔ ⊤ )
5	2 3 4	3bitr4ri	⊢ ( [ 𝑧 / 𝑥 ] ⊤ ↔ 𝑧 ∈ { 𝑦 ∣ ⊤ } )
6	1 5	bitri	⊢ ( 𝑧 ∈ { 𝑥 ∣ ⊤ } ↔ 𝑧 ∈ { 𝑦 ∣ ⊤ } )
7	6	eqriv	⊢ { 𝑥 ∣ ⊤ } = { 𝑦 ∣ ⊤ }