Description: Equality theorem for class substitution. (Contributed by Thierry Arnoux, 4-Sep-2018)
Ref | Expression | ||
---|---|---|---|
Hypotheses | sbceqbidf.1 | ⊢ Ⅎ 𝑥 𝜑 | |
sbceqbidf.2 | ⊢ ( 𝜑 → 𝐴 = 𝐵 ) | ||
sbceqbidf.3 | ⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) ) | ||
Assertion | sbceqbidf | ⊢ ( 𝜑 → ( [ 𝐴 / 𝑥 ] 𝜓 ↔ [ 𝐵 / 𝑥 ] 𝜒 ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbceqbidf.1 | ⊢ Ⅎ 𝑥 𝜑 | |
2 | sbceqbidf.2 | ⊢ ( 𝜑 → 𝐴 = 𝐵 ) | |
3 | sbceqbidf.3 | ⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) ) | |
4 | 1 3 | abbid | ⊢ ( 𝜑 → { 𝑥 ∣ 𝜓 } = { 𝑥 ∣ 𝜒 } ) |
5 | 2 4 | eleq12d | ⊢ ( 𝜑 → ( 𝐴 ∈ { 𝑥 ∣ 𝜓 } ↔ 𝐵 ∈ { 𝑥 ∣ 𝜒 } ) ) |
6 | df-sbc | ⊢ ( [ 𝐴 / 𝑥 ] 𝜓 ↔ 𝐴 ∈ { 𝑥 ∣ 𝜓 } ) | |
7 | df-sbc | ⊢ ( [ 𝐵 / 𝑥 ] 𝜒 ↔ 𝐵 ∈ { 𝑥 ∣ 𝜒 } ) | |
8 | 5 6 7 | 3bitr4g | ⊢ ( 𝜑 → ( [ 𝐴 / 𝑥 ] 𝜓 ↔ [ 𝐵 / 𝑥 ] 𝜒 ) ) |