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Mirrors > Home > MPE Home > Th. List > rexab2 | Structured version Visualization version GIF version |
Description: Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.) Drop ax-8 2112. (Revised by Gino Giotto, 1-Dec-2023.) |
Ref | Expression |
---|---|
ralab2.1 | ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
rexab2 | ⊢ (∃𝑥 ∈ {𝑦 ∣ 𝜑}𝜓 ↔ ∃𝑦(𝜑 ∧ 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rex 3144 | . 2 ⊢ (∃𝑥 ∈ {𝑦 ∣ 𝜑}𝜓 ↔ ∃𝑥(𝑥 ∈ {𝑦 ∣ 𝜑} ∧ 𝜓)) | |
2 | nfsab1 2808 | . . . 4 ⊢ Ⅎ𝑦 𝑥 ∈ {𝑦 ∣ 𝜑} | |
3 | nfv 1911 | . . . 4 ⊢ Ⅎ𝑦𝜓 | |
4 | 2, 3 | nfan 1896 | . . 3 ⊢ Ⅎ𝑦(𝑥 ∈ {𝑦 ∣ 𝜑} ∧ 𝜓) |
5 | nfv 1911 | . . 3 ⊢ Ⅎ𝑥(𝜑 ∧ 𝜒) | |
6 | eleq1ab 2801 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ 𝑦 ∈ {𝑦 ∣ 𝜑})) | |
7 | abid 2803 | . . . . 5 ⊢ (𝑦 ∈ {𝑦 ∣ 𝜑} ↔ 𝜑) | |
8 | 6, 7 | syl6bb 289 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ 𝜑)) |
9 | ralab2.1 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)) | |
10 | 8, 9 | anbi12d 632 | . . 3 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ {𝑦 ∣ 𝜑} ∧ 𝜓) ↔ (𝜑 ∧ 𝜒))) |
11 | 4, 5, 10 | cbvexv1 2358 | . 2 ⊢ (∃𝑥(𝑥 ∈ {𝑦 ∣ 𝜑} ∧ 𝜓) ↔ ∃𝑦(𝜑 ∧ 𝜒)) |
12 | 1, 11 | bitri 277 | 1 ⊢ (∃𝑥 ∈ {𝑦 ∣ 𝜑}𝜓 ↔ ∃𝑦(𝜑 ∧ 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∃wex 1776 ∈ wcel 2110 {cab 2799 ∃wrex 3139 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-10 2141 ax-11 2157 ax-12 2173 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-rex 3144 |
This theorem is referenced by: rexrab2 3692 tmdgsum2 22698 clrellem 39975 brtrclfv2 40065 |
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