Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > sbievw2 | Structured version Visualization version GIF version |
Description: sbievw 2102 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 2105 | . . 3 ⊢ ([𝑦 / 𝑤][𝑤 / 𝑥]𝜑 ↔ [𝑦 / 𝑤][𝑦 / 𝑥]𝜑) | |
2 | sbievw2.1 | . . . . 5 ⊢ (𝑥 = 𝑤 → (𝜑 ↔ 𝜒)) | |
3 | 2 | sbievw 2102 | . . . 4 ⊢ ([𝑤 / 𝑥]𝜑 ↔ 𝜒) |
4 | 3 | sbbii 2080 | . . 3 ⊢ ([𝑦 / 𝑤][𝑤 / 𝑥]𝜑 ↔ [𝑦 / 𝑤]𝜒) |
5 | sbv 2097 | . . 3 ⊢ ([𝑦 / 𝑤][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑) | |
6 | 1, 4, 5 | 3bitr3i 303 | . 2 ⊢ ([𝑦 / 𝑤]𝜒 ↔ [𝑦 / 𝑥]𝜑) |
7 | sbievw2.2 | . . 3 ⊢ (𝑤 = 𝑦 → (𝜒 ↔ 𝜓)) | |
8 | 7 | sbievw 2102 | . 2 ⊢ ([𝑦 / 𝑤]𝜒 ↔ 𝜓) |
9 | 6, 8 | bitr3i 279 | 1 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 [wsb 2068 |
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 1969 ax-7 2014 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1780 df-sb 2069 |
This theorem is referenced by: sbco2vv 2107 equsb3 2108 equsb3r 2109 elsb3 2121 elsb4 2129 eqsb3 2938 clelsb3 2939 sbss 4455 |
Copyright terms: Public domain | W3C validator |