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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 2741 | . . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑦 = 𝐵 ↔ 𝐴 = 𝐵)) | |
| 3 | 2 | rexbidv 3179 | . . . . 5 ⊢ (𝑦 = 𝐴 → (∃𝑥 ∈ 𝐶 𝑦 = 𝐵 ↔ ∃𝑥 ∈ 𝐶 𝐴 = 𝐵)) |
| 4 | 3 | elabg 3676 | . . . 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 476 | . 2 ⊢ ((𝜑 ∧ 𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐶 𝑦 = 𝐵}) → ∃𝑥 ∈ 𝐶 𝐴 = 𝐵) |
| 7 | elabreximd.1 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
| 8 | elabreximd.2 | . . . 4 ⊢ Ⅎ𝑥𝜒 | |
| 9 | simpr 484 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝐴 = 𝐵) → 𝐴 = 𝐵) | |
| 10 | elabreximd.5 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝜓) | |
| 11 | 10 | adantr 480 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝐴 = 𝐵) → 𝜓) |
| 12 | elabreximd.3 | . . . . . . 7 ⊢ (𝐴 = 𝐵 → (𝜒 ↔ 𝜓)) | |
| 13 | 12 | biimpar 477 | . . . . . 6 ⊢ ((𝐴 = 𝐵 ∧ 𝜓) → 𝜒) |
| 14 | 9, 11, 13 | syl2anc 584 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝐴 = 𝐵) → 𝜒) |
| 15 | 14 | exp31 419 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐶 → (𝐴 = 𝐵 → 𝜒))) |
| 16 | 7, 8, 15 | rexlimd 3266 | . . 3 ⊢ (𝜑 → (∃𝑥 ∈ 𝐶 𝐴 = 𝐵 → 𝜒)) |
| 17 | 16 | imp 406 | . 2 ⊢ ((𝜑 ∧ ∃𝑥 ∈ 𝐶 𝐴 = 𝐵) → 𝜒) |
| 18 | 6, 17 | syldan 591 | 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 Ⅎwnf 1783 ∈ 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: elabreximdv 32530 abrexss 32531 iinabrex 32582 disjabrex 32595 disjabrexf 32596 |
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