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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 36902 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 3923 | . . 3 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐴 | |
| 2 | nfcsb1v 3923 | . . 3 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵 | |
| 3 | 1, 2 | nfeq 2919 | . 2 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑦 / 𝑥⦌𝐵 |
| 4 | csbeq1a 3913 | . . 3 ⊢ (𝑥 = 𝑦 → 𝐴 = ⦋𝑦 / 𝑥⦌𝐴) | |
| 5 | csbeq1a 3913 | . . 3 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵) | |
| 6 | 4, 5 | eqeq12d 2753 | . 2 ⊢ (𝑥 = 𝑦 → (𝐴 = 𝐵 ↔ ⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑦 / 𝑥⦌𝐵)) |
| 7 | 3, 6 | sbiev 2314 | 1 ⊢ ([𝑦 / 𝑥]𝐴 = 𝐵 ↔ ⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑦 / 𝑥⦌𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 = wceq 1540 [wsb 2064 ⦋csb 3899 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-sbc 3789 df-csb 3900 |
| This theorem is referenced by: (None) |
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