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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 3554 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 2997 | . . 3 ⊢ Ⅎ𝑥 𝑦 = 𝐴 |
3 | nfcsb1v 3909 | . . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵 | |
4 | csbhypf.2 | . . . 4 ⊢ Ⅎ𝑥𝐶 | |
5 | 3, 4 | nfeq 2993 | . . 3 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵 = 𝐶 |
6 | 2, 5 | nfim 1897 | . 2 ⊢ Ⅎ𝑥(𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐵 = 𝐶) |
7 | eqeq1 2827 | . . 3 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝐴 ↔ 𝑦 = 𝐴)) | |
8 | csbeq1a 3899 | . . . 4 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵) | |
9 | 8 | eqeq1d 2825 | . . 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 2242 | 1 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐵 = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 Ⅎwnfc 2963 ⦋csb 3885 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-sbc 3775 df-csb 3886 |
This theorem is referenced by: disji2 5050 disjprgw 5063 disjprg 5064 disjxun 5066 tfisi 7575 coe1fzgsumdlem 20471 evl1gsumdlem 20521 iundisj2 24152 disji2f 30329 disjif2 30333 iundisj2f 30342 iundisj2fi 30522 |
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