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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > elabreximd | Structured version Visualization version GIF version |
Description: Class substitution in an image set. (Contributed by Thierry Arnoux, 30-Dec-2016.) |
Ref | Expression |
---|---|
elabreximd.1 | ⊢ Ⅎ𝑥𝜑 |
elabreximd.2 | ⊢ Ⅎ𝑥𝜒 |
elabreximd.3 | ⊢ (𝐴 = 𝐵 → (𝜒 ↔ 𝜓)) |
elabreximd.4 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
elabreximd.5 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝜓) |
Ref | Expression |
---|---|
elabreximd | ⊢ ((𝜑 ∧ 𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐶 𝑦 = 𝐵}) → 𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elabreximd.4 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
2 | eqeq1 2782 | . . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑦 = 𝐵 ↔ 𝐴 = 𝐵)) | |
3 | 2 | rexbidv 3237 | . . . . 5 ⊢ (𝑦 = 𝐴 → (∃𝑥 ∈ 𝐶 𝑦 = 𝐵 ↔ ∃𝑥 ∈ 𝐶 𝐴 = 𝐵)) |
4 | 3 | elabg 3556 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐶 𝑦 = 𝐵} ↔ ∃𝑥 ∈ 𝐶 𝐴 = 𝐵)) |
5 | 1, 4 | syl 17 | . . 3 ⊢ (𝜑 → (𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐶 𝑦 = 𝐵} ↔ ∃𝑥 ∈ 𝐶 𝐴 = 𝐵)) |
6 | 5 | biimpa 470 | . 2 ⊢ ((𝜑 ∧ 𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐶 𝑦 = 𝐵}) → ∃𝑥 ∈ 𝐶 𝐴 = 𝐵) |
7 | elabreximd.1 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
8 | elabreximd.2 | . . . 4 ⊢ Ⅎ𝑥𝜒 | |
9 | simpr 479 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝐴 = 𝐵) → 𝐴 = 𝐵) | |
10 | elabreximd.5 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝜓) | |
11 | 10 | adantr 474 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝐴 = 𝐵) → 𝜓) |
12 | elabreximd.3 | . . . . . . 7 ⊢ (𝐴 = 𝐵 → (𝜒 ↔ 𝜓)) | |
13 | 12 | biimpar 471 | . . . . . 6 ⊢ ((𝐴 = 𝐵 ∧ 𝜓) → 𝜒) |
14 | 9, 11, 13 | syl2anc 579 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝐴 = 𝐵) → 𝜒) |
15 | 14 | exp31 412 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐶 → (𝐴 = 𝐵 → 𝜒))) |
16 | 7, 8, 15 | rexlimd 3208 | . . 3 ⊢ (𝜑 → (∃𝑥 ∈ 𝐶 𝐴 = 𝐵 → 𝜒)) |
17 | 16 | imp 397 | . 2 ⊢ ((𝜑 ∧ ∃𝑥 ∈ 𝐶 𝐴 = 𝐵) → 𝜒) |
18 | 6, 17 | syldan 585 | 1 ⊢ ((𝜑 ∧ 𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐶 𝑦 = 𝐵}) → 𝜒) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1601 Ⅎwnf 1827 ∈ wcel 2107 {cab 2763 ∃wrex 3091 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-ext 2754 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ral 3095 df-rex 3096 df-v 3400 |
This theorem is referenced by: elabreximdv 29928 abrexss 29929 disjabrex 29975 disjabrexf 29976 |
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