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| Description: sbievw 2092 applied twice, avoiding a DV condition on 𝑥, 𝑦. Based on proofs by Wolf Lammen. (Contributed by Steven Nguyen, 29-Jul-2023.) | 
| Ref | Expression | 
|---|---|
| sbievw2.1 | ⊢ (𝑥 = 𝑤 → (𝜑 ↔ 𝜒)) | 
| sbievw2.2 | ⊢ (𝑤 = 𝑦 → (𝜒 ↔ 𝜓)) | 
| Ref | Expression | 
|---|---|
| sbievw2 | ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | sbcom3vv 2096 | . . 3 ⊢ ([𝑦 / 𝑤][𝑤 / 𝑥]𝜑 ↔ [𝑦 / 𝑤][𝑦 / 𝑥]𝜑) | |
| 2 | sbievw2.1 | . . . . 5 ⊢ (𝑥 = 𝑤 → (𝜑 ↔ 𝜒)) | |
| 3 | 2 | sbievw 2092 | . . . 4 ⊢ ([𝑤 / 𝑥]𝜑 ↔ 𝜒) | 
| 4 | 3 | sbbii 2075 | . . 3 ⊢ ([𝑦 / 𝑤][𝑤 / 𝑥]𝜑 ↔ [𝑦 / 𝑤]𝜒) | 
| 5 | sbv 2087 | . . 3 ⊢ ([𝑦 / 𝑤][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑) | |
| 6 | 1, 4, 5 | 3bitr3i 301 | . 2 ⊢ ([𝑦 / 𝑤]𝜒 ↔ [𝑦 / 𝑥]𝜑) | 
| 7 | sbievw2.2 | . . 3 ⊢ (𝑤 = 𝑦 → (𝜒 ↔ 𝜓)) | |
| 8 | 7 | sbievw 2092 | . 2 ⊢ ([𝑦 / 𝑤]𝜒 ↔ 𝜓) | 
| 9 | 6, 8 | bitr3i 277 | 1 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 [wsb 2063 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1779 df-sb 2064 | 
| This theorem is referenced by: sbco2vv 2098 equsb3 2102 equsb3r 2103 elsb1 2115 elsb2 2124 eqsb1 2866 clelsb1 2867 clelsb2 2868 sbss 4518 | 
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