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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 2802 | . . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑦 = 𝐵 ↔ 𝐴 = 𝐵)) | |
3 | 2 | rexbidv 3256 | . . . . 5 ⊢ (𝑦 = 𝐴 → (∃𝑥 ∈ 𝐶 𝑦 = 𝐵 ↔ ∃𝑥 ∈ 𝐶 𝐴 = 𝐵)) |
4 | 3 | elabg 3614 | . . . 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 480 | . 2 ⊢ ((𝜑 ∧ 𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐶 𝑦 = 𝐵}) → ∃𝑥 ∈ 𝐶 𝐴 = 𝐵) |
7 | elabreximd.1 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
8 | elabreximd.2 | . . . 4 ⊢ Ⅎ𝑥𝜒 | |
9 | simpr 488 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝐴 = 𝐵) → 𝐴 = 𝐵) | |
10 | elabreximd.5 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝜓) | |
11 | 10 | adantr 484 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝐴 = 𝐵) → 𝜓) |
12 | elabreximd.3 | . . . . . . 7 ⊢ (𝐴 = 𝐵 → (𝜒 ↔ 𝜓)) | |
13 | 12 | biimpar 481 | . . . . . 6 ⊢ ((𝐴 = 𝐵 ∧ 𝜓) → 𝜒) |
14 | 9, 11, 13 | syl2anc 587 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝐴 = 𝐵) → 𝜒) |
15 | 14 | exp31 423 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐶 → (𝐴 = 𝐵 → 𝜒))) |
16 | 7, 8, 15 | rexlimd 3276 | . . 3 ⊢ (𝜑 → (∃𝑥 ∈ 𝐶 𝐴 = 𝐵 → 𝜒)) |
17 | 16 | imp 410 | . 2 ⊢ ((𝜑 ∧ ∃𝑥 ∈ 𝐶 𝐴 = 𝐵) → 𝜒) |
18 | 6, 17 | syldan 594 | 1 ⊢ ((𝜑 ∧ 𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐶 𝑦 = 𝐵}) → 𝜒) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 Ⅎwnf 1785 ∈ wcel 2111 {cab 2776 ∃wrex 3107 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 |
This theorem is referenced by: elabreximdv 30279 abrexss 30280 iinabrex 30332 disjabrex 30345 disjabrexf 30346 |
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