Theorem bj-cleq

Description: Substitution property for certain classes. (Contributed by BJ, 2-Apr-2019)

	Ref	Expression
Assertion	bj-cleq	⊢ ( 𝐴 = 𝐵 → { 𝑥 ∣ { 𝑥 } ∈ ( 𝐴 “ 𝐶 ) } = { 𝑥 ∣ { 𝑥 } ∈ ( 𝐵 “ 𝐶 ) } )

Proof

Step	Hyp	Ref	Expression
1		imaeq1	⊢ ( 𝐴 = 𝐵 → ( 𝐴 “ 𝐶 ) = ( 𝐵 “ 𝐶 ) )
2		eleq2	⊢ ( ( 𝐴 “ 𝐶 ) = ( 𝐵 “ 𝐶 ) → ( { 𝑥 } ∈ ( 𝐴 “ 𝐶 ) ↔ { 𝑥 } ∈ ( 𝐵 “ 𝐶 ) ) )
3	2	alrimiv	⊢ ( ( 𝐴 “ 𝐶 ) = ( 𝐵 “ 𝐶 ) → ∀ 𝑥 ( { 𝑥 } ∈ ( 𝐴 “ 𝐶 ) ↔ { 𝑥 } ∈ ( 𝐵 “ 𝐶 ) ) )
4		abbi	⊢ ( ∀ 𝑥 ( { 𝑥 } ∈ ( 𝐴 “ 𝐶 ) ↔ { 𝑥 } ∈ ( 𝐵 “ 𝐶 ) ) → { 𝑥 ∣ { 𝑥 } ∈ ( 𝐴 “ 𝐶 ) } = { 𝑥 ∣ { 𝑥 } ∈ ( 𝐵 “ 𝐶 ) } )
5	1 3 4	3syl	⊢ ( 𝐴 = 𝐵 → { 𝑥 ∣ { 𝑥 } ∈ ( 𝐴 “ 𝐶 ) } = { 𝑥 ∣ { 𝑥 } ∈ ( 𝐵 “ 𝐶 ) } )