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| Mirrors > Home > MPE Home > Th. List > Mathboxes > wl-2spsbbi | Structured version Visualization version GIF version | ||
| Description: spsbbi 2072 applied twice. (Contributed by Wolf Lammen, 5-Aug-2023.) | 
| Ref | Expression | 
|---|---|
| wl-2spsbbi | ⊢ (∀𝑎∀𝑏(𝜑 ↔ 𝜓) → ([𝑦 / 𝑏][𝑥 / 𝑎]𝜑 ↔ [𝑦 / 𝑏][𝑥 / 𝑎]𝜓)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | alcom 2158 | . 2 ⊢ (∀𝑎∀𝑏(𝜑 ↔ 𝜓) ↔ ∀𝑏∀𝑎(𝜑 ↔ 𝜓)) | |
| 2 | nfa1 2150 | . . 3 ⊢ Ⅎ𝑏∀𝑏∀𝑎(𝜑 ↔ 𝜓) | |
| 3 | nfa1 2150 | . . . . 5 ⊢ Ⅎ𝑎∀𝑎(𝜑 ↔ 𝜓) | |
| 4 | sp 2182 | . . . . 5 ⊢ (∀𝑎(𝜑 ↔ 𝜓) → (𝜑 ↔ 𝜓)) | |
| 5 | 3, 4 | sbbid 2245 | . . . 4 ⊢ (∀𝑎(𝜑 ↔ 𝜓) → ([𝑥 / 𝑎]𝜑 ↔ [𝑥 / 𝑎]𝜓)) | 
| 6 | 5 | sps 2184 | . . 3 ⊢ (∀𝑏∀𝑎(𝜑 ↔ 𝜓) → ([𝑥 / 𝑎]𝜑 ↔ [𝑥 / 𝑎]𝜓)) | 
| 7 | 2, 6 | sbbid 2245 | . 2 ⊢ (∀𝑏∀𝑎(𝜑 ↔ 𝜓) → ([𝑦 / 𝑏][𝑥 / 𝑎]𝜑 ↔ [𝑦 / 𝑏][𝑥 / 𝑎]𝜓)) | 
| 8 | 1, 7 | sylbi 217 | 1 ⊢ (∀𝑎∀𝑏(𝜑 ↔ 𝜓) → ([𝑦 / 𝑏][𝑥 / 𝑎]𝜑 ↔ [𝑦 / 𝑏][𝑥 / 𝑎]𝜓)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∀wal 1537 [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 ax-10 2140 ax-11 2156 ax-12 2176 | 
| This theorem depends on definitions: df-bi 207 df-or 848 df-ex 1779 df-nf 1783 df-sb 2064 | 
| This theorem is referenced by: (None) | 
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