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Mirrors > Home > MPE Home > Th. List > sbco2 | Structured version Visualization version GIF version |
Description: A composition law for substitution. (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 17-Sep-2018.) |
Ref | Expression |
---|---|
sbco2.1 | ⊢ Ⅎ𝑧𝜑 |
Ref | Expression |
---|---|
sbco2 | ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbequ12 2267 | . . . 4 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑧][𝑧 / 𝑥]𝜑)) | |
2 | sbequ 2523 | . . . 4 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)) | |
3 | 1, 2 | bitr3d 270 | . . 3 ⊢ (𝑧 = 𝑦 → ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)) |
4 | 3 | sps 2209 | . 2 ⊢ (∀𝑧 𝑧 = 𝑦 → ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)) |
5 | nfnae 2470 | . . 3 ⊢ Ⅎ𝑧 ¬ ∀𝑧 𝑧 = 𝑦 | |
6 | sbco2.1 | . . . 4 ⊢ Ⅎ𝑧𝜑 | |
7 | 6 | nfsb4 2537 | . . 3 ⊢ (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧[𝑦 / 𝑥]𝜑) |
8 | 2 | a1i 11 | . . 3 ⊢ (¬ ∀𝑧 𝑧 = 𝑦 → (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))) |
9 | 5, 7, 8 | sbied 2556 | . 2 ⊢ (¬ ∀𝑧 𝑧 = 𝑦 → ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)) |
10 | 4, 9 | pm2.61i 176 | 1 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∀wal 1629 Ⅎwnf 1856 [wsb 2049 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 829 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 |
This theorem is referenced by: sbco2d 2563 equsb3ALT 2581 elsb3OLD 2584 elsb4OLD 2587 sb7f 2601 sbco4lem 2613 sbco4 2614 eqsb3 2877 clelsb3 2878 cbvab 2895 clelsb3f 2917 sbralie 3333 sbcco 3611 |
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