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Mathbox for Wolf Lammen |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > wl-2spsbbi | Structured version Visualization version GIF version |
Description: spsbbi 2071 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 2157 | . 2 ⊢ (∀𝑎∀𝑏(𝜑 ↔ 𝜓) ↔ ∀𝑏∀𝑎(𝜑 ↔ 𝜓)) | |
2 | nfa1 2149 | . . 3 ⊢ Ⅎ𝑏∀𝑏∀𝑎(𝜑 ↔ 𝜓) | |
3 | nfa1 2149 | . . . . 5 ⊢ Ⅎ𝑎∀𝑎(𝜑 ↔ 𝜓) | |
4 | sp 2181 | . . . . 5 ⊢ (∀𝑎(𝜑 ↔ 𝜓) → (𝜑 ↔ 𝜓)) | |
5 | 3, 4 | sbbid 2244 | . . . 4 ⊢ (∀𝑎(𝜑 ↔ 𝜓) → ([𝑥 / 𝑎]𝜑 ↔ [𝑥 / 𝑎]𝜓)) |
6 | 5 | sps 2183 | . . 3 ⊢ (∀𝑏∀𝑎(𝜑 ↔ 𝜓) → ([𝑥 / 𝑎]𝜑 ↔ [𝑥 / 𝑎]𝜓)) |
7 | 2, 6 | sbbid 2244 | . 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 1535 [wsb 2062 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-10 2139 ax-11 2155 ax-12 2175 |
This theorem depends on definitions: df-bi 207 df-or 848 df-ex 1777 df-nf 1781 df-sb 2063 |
This theorem is referenced by: (None) |
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