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Mirrors > Home > NFE Home > Th. List > sbco2 | GIF version |
Description: A composition law for substitution. (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 6-Oct-2016.) |
Ref | Expression |
---|---|
sbco2.1 | ⊢ Ⅎzφ |
Ref | Expression |
---|---|
sbco2 | ⊢ ([y / z][z / x]φ ↔ [y / x]φ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbco2.1 | . . . . . 6 ⊢ Ⅎzφ | |
2 | 1 | sbid2 2084 | . . . . 5 ⊢ ([x / z][z / x]φ ↔ φ) |
3 | sbequ 2060 | . . . . 5 ⊢ (x = y → ([x / z][z / x]φ ↔ [y / z][z / x]φ)) | |
4 | 2, 3 | syl5bbr 250 | . . . 4 ⊢ (x = y → (φ ↔ [y / z][z / x]φ)) |
5 | sbequ12 1919 | . . . 4 ⊢ (x = y → (φ ↔ [y / x]φ)) | |
6 | 4, 5 | bitr3d 246 | . . 3 ⊢ (x = y → ([y / z][z / x]φ ↔ [y / x]φ)) |
7 | 6 | sps 1754 | . 2 ⊢ (∀x x = y → ([y / z][z / x]φ ↔ [y / x]φ)) |
8 | nfnae 1956 | . . . 4 ⊢ Ⅎx ¬ ∀x x = y | |
9 | 1 | nfs1 2044 | . . . . 5 ⊢ Ⅎx[z / x]φ |
10 | 9 | nfsb4 2081 | . . . 4 ⊢ (¬ ∀x x = y → Ⅎx[y / z][z / x]φ) |
11 | 4 | a1i 10 | . . . 4 ⊢ (¬ ∀x x = y → (x = y → (φ ↔ [y / z][z / x]φ))) |
12 | 8, 10, 11 | sbied 2036 | . . 3 ⊢ (¬ ∀x x = y → ([y / x]φ ↔ [y / z][z / x]φ)) |
13 | 12 | bicomd 192 | . 2 ⊢ (¬ ∀x x = y → ([y / z][z / x]φ ↔ [y / x]φ)) |
14 | 7, 13 | pm2.61i 156 | 1 ⊢ ([y / z][z / x]φ ↔ [y / x]φ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 176 ∀wal 1540 Ⅎwnf 1544 [wsb 1648 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 |
This theorem is referenced by: sbco2d 2087 equsb3 2102 elsb3 2103 elsb4 2104 dfsb7 2119 sb7f 2120 2eu6 2289 eqsb3 2454 clelsb3 2455 sbralie 2848 sbcco 3068 |
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