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Mirrors > Home > NFE Home > Th. List > csbhypf | GIF version |
Description: Introduce an explicit substitution into an implicit substitution hypothesis. See sbhypf 2904 for class substitution version. (Contributed by NM, 19-Dec-2008.) |
Ref | Expression |
---|---|
csbhypf.1 | ⊢ ℲxA |
csbhypf.2 | ⊢ ℲxC |
csbhypf.3 | ⊢ (x = A → B = C) |
Ref | Expression |
---|---|
csbhypf | ⊢ (y = A → [y / x]B = C) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | csbhypf.1 | . . . 4 ⊢ ℲxA | |
2 | 1 | nfeq2 2500 | . . 3 ⊢ Ⅎx y = A |
3 | nfcsb1v 3168 | . . . 4 ⊢ Ⅎx[y / x]B | |
4 | csbhypf.2 | . . . 4 ⊢ ℲxC | |
5 | 3, 4 | nfeq 2496 | . . 3 ⊢ Ⅎx[y / x]B = C |
6 | 2, 5 | nfim 1813 | . 2 ⊢ Ⅎx(y = A → [y / x]B = C) |
7 | eqeq1 2359 | . . 3 ⊢ (x = y → (x = A ↔ y = A)) | |
8 | csbeq1a 3144 | . . . 4 ⊢ (x = y → B = [y / x]B) | |
9 | 8 | eqeq1d 2361 | . . 3 ⊢ (x = y → (B = C ↔ [y / x]B = C)) |
10 | 7, 9 | imbi12d 311 | . 2 ⊢ (x = y → ((x = A → B = C) ↔ (y = A → [y / x]B = C))) |
11 | csbhypf.3 | . 2 ⊢ (x = A → B = C) | |
12 | 6, 10, 11 | chvar 1986 | 1 ⊢ (y = A → [y / x]B = C) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1642 Ⅎwnfc 2476 [csb 3136 |
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-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-sbc 3047 df-csb 3137 |
This theorem is referenced by: (None) |
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