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Mirrors > Home > MPE Home > Th. List > sbievw2 | Structured version Visualization version GIF version |
Description: sbievw 2094 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 2097 | . . 3 ⊢ ([𝑦 / 𝑤][𝑤 / 𝑥]𝜑 ↔ [𝑦 / 𝑤][𝑦 / 𝑥]𝜑) | |
2 | sbievw2.1 | . . . . 5 ⊢ (𝑥 = 𝑤 → (𝜑 ↔ 𝜒)) | |
3 | 2 | sbievw 2094 | . . . 4 ⊢ ([𝑤 / 𝑥]𝜑 ↔ 𝜒) |
4 | 3 | sbbii 2078 | . . 3 ⊢ ([𝑦 / 𝑤][𝑤 / 𝑥]𝜑 ↔ [𝑦 / 𝑤]𝜒) |
5 | sbv 2090 | . . 3 ⊢ ([𝑦 / 𝑤][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑) | |
6 | 1, 4, 5 | 3bitr3i 300 | . 2 ⊢ ([𝑦 / 𝑤]𝜒 ↔ [𝑦 / 𝑥]𝜑) |
7 | sbievw2.2 | . . 3 ⊢ (𝑤 = 𝑦 → (𝜒 ↔ 𝜓)) | |
8 | 7 | sbievw 2094 | . 2 ⊢ ([𝑦 / 𝑤]𝜒 ↔ 𝜓) |
9 | 6, 8 | bitr3i 276 | 1 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 [wsb 2066 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 |
This theorem depends on definitions: df-bi 206 df-an 397 df-ex 1781 df-sb 2067 |
This theorem is referenced by: sbco2vv 2099 equsb3 2100 equsb3r 2101 elsb1 2113 elsb2 2122 eqsb1 2863 clelsb1 2864 clelsb2 2865 sbss 4467 |
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