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