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Mirrors > Home > MPE Home > Th. List > vtoclgaf | Structured version Visualization version GIF version |
Description: Implicit substitution of a class for a setvar variable. (Contributed by NM, 17-Feb-2006.) (Revised by Mario Carneiro, 10-Oct-2016.) |
Ref | Expression |
---|---|
vtoclgaf.1 | ⊢ Ⅎ𝑥𝐴 |
vtoclgaf.2 | ⊢ Ⅎ𝑥𝜓 |
vtoclgaf.3 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
vtoclgaf.4 | ⊢ (𝑥 ∈ 𝐵 → 𝜑) |
Ref | Expression |
---|---|
vtoclgaf | ⊢ (𝐴 ∈ 𝐵 → 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vtoclgaf.1 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
2 | 1 | nfel1 2996 | . . . 4 ⊢ Ⅎ𝑥 𝐴 ∈ 𝐵 |
3 | vtoclgaf.2 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
4 | 2, 3 | nfim 1897 | . . 3 ⊢ Ⅎ𝑥(𝐴 ∈ 𝐵 → 𝜓) |
5 | eleq1 2902 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵)) | |
6 | vtoclgaf.3 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
7 | 5, 6 | imbi12d 347 | . . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ 𝐵 → 𝜑) ↔ (𝐴 ∈ 𝐵 → 𝜓))) |
8 | vtoclgaf.4 | . . 3 ⊢ (𝑥 ∈ 𝐵 → 𝜑) | |
9 | 1, 4, 7, 8 | vtoclgf 3567 | . 2 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ 𝐵 → 𝜓)) |
10 | 9 | pm2.43i 52 | 1 ⊢ (𝐴 ∈ 𝐵 → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1537 Ⅎwnf 1784 ∈ wcel 2114 Ⅎwnfc 2963 |
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-v 3498 |
This theorem is referenced by: ssiun2s 4974 iunopeqop 5413 fvmptss 6782 fvmptf 6791 fmptco 6893 tfis 7571 inar1 10199 sumss 15083 fprodn0 15335 prmind2 16031 lss1d 19737 itg2splitlem 24351 dgrle 24835 cnlnadjlem5 29850 poimirlem25 34919 stoweidlem26 42318 |
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