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 ) } ) |
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 | abbi1 | ⊢ ( ∀ 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 ) } ) |