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Mirrors > Home > MPE Home > Th. List > csbhypf | Structured version Visualization version GIF version |
Description: Introduce an explicit substitution into an implicit substitution hypothesis. See sbhypf 3552 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 2995 | . . 3 ⊢ Ⅎ𝑥 𝑦 = 𝐴 |
3 | nfcsb1v 3906 | . . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵 | |
4 | csbhypf.2 | . . . 4 ⊢ Ⅎ𝑥𝐶 | |
5 | 3, 4 | nfeq 2991 | . . 3 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵 = 𝐶 |
6 | 2, 5 | nfim 1893 | . 2 ⊢ Ⅎ𝑥(𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐵 = 𝐶) |
7 | eqeq1 2825 | . . 3 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝐴 ↔ 𝑦 = 𝐴)) | |
8 | csbeq1a 3896 | . . . 4 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵) | |
9 | 8 | eqeq1d 2823 | . . 3 ⊢ (𝑥 = 𝑦 → (𝐵 = 𝐶 ↔ ⦋𝑦 / 𝑥⦌𝐵 = 𝐶)) |
10 | 7, 9 | imbi12d 347 | . 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 | chvarfv 2238 | 1 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐵 = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 Ⅎwnfc 2961 ⦋csb 3882 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-sbc 3772 df-csb 3883 |
This theorem is referenced by: disji2 5047 disjprgw 5060 disjprg 5061 disjxun 5063 tfisi 7572 coe1fzgsumdlem 20468 evl1gsumdlem 20518 iundisj2 24149 disji2f 30326 disjif2 30330 iundisj2f 30339 iundisj2fi 30519 |
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