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Mirrors > Home > NFE Home > Th. List > csbiotag | GIF version |
Description: Class substitution within a description binder. (Contributed by Scott Fenton, 6-Oct-2017.) |
Ref | Expression |
---|---|
csbiotag | ⊢ (A ∈ V → [A / x](℩yφ) = (℩y[̣A / x]̣φ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | csbeq1 3139 | . . 3 ⊢ (z = A → [z / x](℩yφ) = [A / x](℩yφ)) | |
2 | dfsbcq2 3049 | . . . 4 ⊢ (z = A → ([z / x]φ ↔ [̣A / x]̣φ)) | |
3 | 2 | iotabidv 4360 | . . 3 ⊢ (z = A → (℩y[z / x]φ) = (℩y[̣A / x]̣φ)) |
4 | 1, 3 | eqeq12d 2367 | . 2 ⊢ (z = A → ([z / x](℩yφ) = (℩y[z / x]φ) ↔ [A / x](℩yφ) = (℩y[̣A / x]̣φ))) |
5 | vex 2862 | . . 3 ⊢ z ∈ V | |
6 | nfs1v 2106 | . . . 4 ⊢ Ⅎx[z / x]φ | |
7 | 6 | nfiota 4343 | . . 3 ⊢ Ⅎx(℩y[z / x]φ) |
8 | sbequ12 1919 | . . . 4 ⊢ (x = z → (φ ↔ [z / x]φ)) | |
9 | 8 | iotabidv 4360 | . . 3 ⊢ (x = z → (℩yφ) = (℩y[z / x]φ)) |
10 | 5, 7, 9 | csbief 3177 | . 2 ⊢ [z / x](℩yφ) = (℩y[z / x]φ) |
11 | 4, 10 | vtoclg 2914 | 1 ⊢ (A ∈ V → [A / x](℩yφ) = (℩y[̣A / x]̣φ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1642 [wsb 1648 ∈ wcel 1710 [̣wsbc 3046 [csb 3136 ℩cio 4337 |
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 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3an 936 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ral 2619 df-rex 2620 df-v 2861 df-sbc 3047 df-csb 3137 df-sn 3741 df-uni 3892 df-iota 4339 |
This theorem is referenced by: csbfv12g 5336 |
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