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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 2824 | . . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑦 = 𝐵 ↔ 𝐴 = 𝐵)) | |
3 | 2 | rexbidv 3296 | . . . . 5 ⊢ (𝑦 = 𝐴 → (∃𝑥 ∈ 𝐶 𝑦 = 𝐵 ↔ ∃𝑥 ∈ 𝐶 𝐴 = 𝐵)) |
4 | 3 | elabg 3662 | . . . 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 479 | . 2 ⊢ ((𝜑 ∧ 𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐶 𝑦 = 𝐵}) → ∃𝑥 ∈ 𝐶 𝐴 = 𝐵) |
7 | elabreximd.1 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
8 | elabreximd.2 | . . . 4 ⊢ Ⅎ𝑥𝜒 | |
9 | simpr 487 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝐴 = 𝐵) → 𝐴 = 𝐵) | |
10 | elabreximd.5 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝜓) | |
11 | 10 | adantr 483 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝐴 = 𝐵) → 𝜓) |
12 | elabreximd.3 | . . . . . . 7 ⊢ (𝐴 = 𝐵 → (𝜒 ↔ 𝜓)) | |
13 | 12 | biimpar 480 | . . . . . 6 ⊢ ((𝐴 = 𝐵 ∧ 𝜓) → 𝜒) |
14 | 9, 11, 13 | syl2anc 586 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝐴 = 𝐵) → 𝜒) |
15 | 14 | exp31 422 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐶 → (𝐴 = 𝐵 → 𝜒))) |
16 | 7, 8, 15 | rexlimd 3316 | . . 3 ⊢ (𝜑 → (∃𝑥 ∈ 𝐶 𝐴 = 𝐵 → 𝜒)) |
17 | 16 | imp 409 | . 2 ⊢ ((𝜑 ∧ ∃𝑥 ∈ 𝐶 𝐴 = 𝐵) → 𝜒) |
18 | 6, 17 | syldan 593 | 1 ⊢ ((𝜑 ∧ 𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐶 𝑦 = 𝐵}) → 𝜒) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1536 Ⅎwnf 1783 ∈ wcel 2113 {cab 2798 ∃wrex 3138 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ral 3142 df-rex 3143 |
This theorem is referenced by: elabreximdv 30270 abrexss 30271 disjabrex 30332 disjabrexf 30333 |
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