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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-sbeqALT | Structured version Visualization version GIF version |
Description: Substitution in an equality (use the more general version bj-sbeq 34213 instead, without disjoint variable condition). (Contributed by BJ, 6-Oct-2018.) (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-sbeqALT | ⊢ ([𝑦 / 𝑥]𝐴 = 𝐵 ↔ ⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑦 / 𝑥⦌𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcsb1v 3907 | . . 3 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐴 | |
2 | nfcsb1v 3907 | . . 3 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵 | |
3 | 1, 2 | nfeq 2991 | . 2 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑦 / 𝑥⦌𝐵 |
4 | csbeq1a 3897 | . . 3 ⊢ (𝑥 = 𝑦 → 𝐴 = ⦋𝑦 / 𝑥⦌𝐴) | |
5 | csbeq1a 3897 | . . 3 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵) | |
6 | 4, 5 | eqeq12d 2837 | . 2 ⊢ (𝑥 = 𝑦 → (𝐴 = 𝐵 ↔ ⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑦 / 𝑥⦌𝐵)) |
7 | 3, 6 | sbiev 2326 | 1 ⊢ ([𝑦 / 𝑥]𝐴 = 𝐵 ↔ ⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑦 / 𝑥⦌𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1533 [wsb 2065 ⦋csb 3883 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-sbc 3773 df-csb 3884 |
This theorem is referenced by: (None) |
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