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| Mirrors > Home > MPE Home > Th. List > Mathboxes > elabreximdv | Structured version Visualization version GIF version | ||
| Description: Class substitution in an image set. (Contributed by Thierry Arnoux, 30-Dec-2016.) |
| Ref | Expression |
|---|---|
| elabreximdv.1 | ⊢ (𝐴 = 𝐵 → (𝜒 ↔ 𝜓)) |
| elabreximdv.2 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
| elabreximdv.3 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝜓) |
| Ref | Expression |
|---|---|
| elabreximdv | ⊢ ((𝜑 ∧ 𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐶 𝑦 = 𝐵}) → 𝜒) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | nfv 1914 | . 2 ⊢ Ⅎ𝑥𝜑 | |
| 2 | nfv 1914 | . 2 ⊢ Ⅎ𝑥𝜒 | |
| 3 | elabreximdv.1 | . 2 ⊢ (𝐴 = 𝐵 → (𝜒 ↔ 𝜓)) | |
| 4 | elabreximdv.2 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
| 5 | elabreximdv.3 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝜓) | |
| 6 | 1, 2, 3, 4, 5 | elabreximd 32529 | 1 ⊢ ((𝜑 ∧ 𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐶 𝑦 = 𝐵}) → 𝜒) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 {cab 2714 ∃wrex 3070 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-12 2177 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 |
| This theorem is referenced by: (None) |
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