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Mirrors > Home > NFE Home > Th. List > csbovg | GIF version |
Description: Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.) (Proof shortened by Mario Carneiro, 5-Dec-2016.) |
Ref | Expression |
---|---|
csbovg | ⊢ (A ∈ D → [A / x](BFC) = ([A / x]B[A / x]F[A / x]C)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | csbeq1 3139 | . . 3 ⊢ (y = A → [y / x](BFC) = [A / x](BFC)) | |
2 | csbeq1 3139 | . . . 4 ⊢ (y = A → [y / x]F = [A / x]F) | |
3 | csbeq1 3139 | . . . 4 ⊢ (y = A → [y / x]B = [A / x]B) | |
4 | csbeq1 3139 | . . . 4 ⊢ (y = A → [y / x]C = [A / x]C) | |
5 | 2, 3, 4 | oveq123d 5543 | . . 3 ⊢ (y = A → ([y / x]B[y / x]F[y / x]C) = ([A / x]B[A / x]F[A / x]C)) |
6 | 1, 5 | eqeq12d 2367 | . 2 ⊢ (y = A → ([y / x](BFC) = ([y / x]B[y / x]F[y / x]C) ↔ [A / x](BFC) = ([A / x]B[A / x]F[A / x]C))) |
7 | vex 2862 | . . 3 ⊢ y ∈ V | |
8 | nfcsb1v 3168 | . . . 4 ⊢ Ⅎx[y / x]B | |
9 | nfcsb1v 3168 | . . . 4 ⊢ Ⅎx[y / x]F | |
10 | nfcsb1v 3168 | . . . 4 ⊢ Ⅎx[y / x]C | |
11 | 8, 9, 10 | nfov 5545 | . . 3 ⊢ Ⅎx([y / x]B[y / x]F[y / x]C) |
12 | csbeq1a 3144 | . . . 4 ⊢ (x = y → F = [y / x]F) | |
13 | csbeq1a 3144 | . . . 4 ⊢ (x = y → B = [y / x]B) | |
14 | csbeq1a 3144 | . . . 4 ⊢ (x = y → C = [y / x]C) | |
15 | 12, 13, 14 | oveq123d 5543 | . . 3 ⊢ (x = y → (BFC) = ([y / x]B[y / x]F[y / x]C)) |
16 | 7, 11, 15 | csbief 3177 | . 2 ⊢ [y / x](BFC) = ([y / x]B[y / x]F[y / x]C) |
17 | 6, 16 | vtoclg 2914 | 1 ⊢ (A ∈ D → [A / x](BFC) = ([A / x]B[A / x]F[A / x]C)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1642 ∈ wcel 1710 [csb 3136 (class class class)co 5525 |
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 ax-nin 4078 ax-xp 4079 ax-cnv 4080 ax-1c 4081 ax-sset 4082 ax-si 4083 ax-ins2 4084 ax-ins3 4085 ax-typlower 4086 ax-sn 4087 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3an 936 df-nan 1288 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-ne 2518 df-ral 2619 df-rex 2620 df-v 2861 df-sbc 3047 df-csb 3137 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-symdif 3216 df-ss 3259 df-nul 3551 df-if 3663 df-pw 3724 df-sn 3741 df-pr 3742 df-uni 3892 df-int 3927 df-opk 4058 df-1c 4136 df-pw1 4137 df-uni1 4138 df-xpk 4185 df-cnvk 4186 df-ins2k 4187 df-ins3k 4188 df-imak 4189 df-cok 4190 df-p6 4191 df-sik 4192 df-ssetk 4193 df-imagek 4194 df-idk 4195 df-iota 4339 df-addc 4378 df-nnc 4379 df-phi 4565 df-op 4566 df-br 4640 df-fv 4795 df-ov 5526 |
This theorem is referenced by: csbov12g 5553 |
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