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Mathbox for Thierry Arnoux |
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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 1911 | . 2 ⊢ Ⅎ𝑥𝜑 | |
2 | nfv 1911 | . 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 32537 | 1 ⊢ ((𝜑 ∧ 𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐶 𝑦 = 𝐵}) → 𝜒) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1536 ∈ wcel 2105 {cab 2711 ∃wrex 3067 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-12 2174 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1539 df-ex 1776 df-nf 1780 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ral 3059 df-rex 3068 |
This theorem is referenced by: (None) |
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