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Mirrors > Home > MPE Home > Th. List > 2sbbii | Structured version Visualization version GIF version |
Description: Infer double substitution into both sides of a logical equivalence. (Contributed by AV, 30-Jul-2023.) |
Ref | Expression |
---|---|
sbbii.1 | ⊢ (𝜑 ↔ 𝜓) |
Ref | Expression |
---|---|
2sbbii | ⊢ ([𝑡 / 𝑥][𝑢 / 𝑦]𝜑 ↔ [𝑡 / 𝑥][𝑢 / 𝑦]𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbbii.1 | . . 3 ⊢ (𝜑 ↔ 𝜓) | |
2 | 1 | sbbii 2077 | . 2 ⊢ ([𝑢 / 𝑦]𝜑 ↔ [𝑢 / 𝑦]𝜓) |
3 | 2 | sbbii 2077 | 1 ⊢ ([𝑡 / 𝑥][𝑢 / 𝑦]𝜑 ↔ [𝑡 / 𝑥][𝑢 / 𝑦]𝜓) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 [wsb 2065 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 |
This theorem depends on definitions: df-bi 206 df-sb 2066 |
This theorem is referenced by: sbco4lem 2270 sbco4lemOLD 2271 ichcircshi 46420 ichbi12i 46426 icheq 46428 ichal 46432 |
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