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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 2765 | . . . 4 ⊢ Ⅎ𝑥 𝐴 ∈ 𝐵 |
3 | vtoclgaf.2 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
4 | 2, 3 | nfim 1813 | . . 3 ⊢ Ⅎ𝑥(𝐴 ∈ 𝐵 → 𝜓) |
5 | eleq1 2676 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵)) | |
6 | vtoclgaf.3 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
7 | 5, 6 | imbi12d 333 | . . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ 𝐵 → 𝜑) ↔ (𝐴 ∈ 𝐵 → 𝜓))) |
8 | vtoclgaf.4 | . . 3 ⊢ (𝑥 ∈ 𝐵 → 𝜑) | |
9 | 1, 4, 7, 8 | vtoclgf 3237 | . 2 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ 𝐵 → 𝜓)) |
10 | 9 | pm2.43i 50 | 1 ⊢ (𝐴 ∈ 𝐵 → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 = wceq 1475 Ⅎwnf 1699 ∈ wcel 1977 Ⅎwnfc 2738 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 |
This theorem is referenced by: vtoclga 3245 ssiun2s 4500 iunopeqop 4906 fvmptss 6201 fvmptf 6209 fmptco 6303 tfis 6946 inar1 9476 sumss 14302 fprodn0 14548 prmind2 15236 lss1d 18784 itg2splitlem 23321 dgrle 23803 cnlnadjlem5 28314 poimirlem25 32604 stoweidlem26 38919 |
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