Description: Necessary deduction regarding substitution of value in equality. (Contributed by RP, 24-Dec-2019)
Ref | Expression | ||
---|---|---|---|
Assertion | frege55lem1c | ⊢ ( ( 𝜑 → [ 𝐴 / 𝑥 ] 𝑥 = 𝐵 ) → ( 𝜑 → 𝐴 = 𝐵 ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-sbc | ⊢ ( [ 𝐴 / 𝑥 ] 𝑥 = 𝐵 ↔ 𝐴 ∈ { 𝑥 ∣ 𝑥 = 𝐵 } ) | |
2 | eqeq1 | ⊢ ( 𝑥 = 𝐴 → ( 𝑥 = 𝐵 ↔ 𝐴 = 𝐵 ) ) | |
3 | 2 | elabg | ⊢ ( 𝐴 ∈ { 𝑥 ∣ 𝑥 = 𝐵 } → ( 𝐴 ∈ { 𝑥 ∣ 𝑥 = 𝐵 } ↔ 𝐴 = 𝐵 ) ) |
4 | 3 | ibi | ⊢ ( 𝐴 ∈ { 𝑥 ∣ 𝑥 = 𝐵 } → 𝐴 = 𝐵 ) |
5 | 1 4 | sylbi | ⊢ ( [ 𝐴 / 𝑥 ] 𝑥 = 𝐵 → 𝐴 = 𝐵 ) |
6 | 5 | imim2i | ⊢ ( ( 𝜑 → [ 𝐴 / 𝑥 ] 𝑥 = 𝐵 ) → ( 𝜑 → 𝐴 = 𝐵 ) ) |