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Mirrors > Home > MPE Home > Th. List > rexab | Structured version Visualization version GIF version |
Description: Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 23-Jan-2014.) (Revised by Mario Carneiro, 3-Sep-2015.) Reduce axiom usage. (Revised by Gino Giotto, 2-Nov-2024.) |
Ref | Expression |
---|---|
ralab.1 | ⊢ (𝑦 = 𝑥 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
rexab | ⊢ (∃𝑥 ∈ {𝑦 ∣ 𝜑}𝜒 ↔ ∃𝑥(𝜓 ∧ 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfrex2 3073 | . . . 4 ⊢ (∃𝑥 ∈ {𝑦 ∣ 𝜑}𝜒 ↔ ¬ ∀𝑥 ∈ {𝑦 ∣ 𝜑} ¬ 𝜒) | |
2 | ralab.1 | . . . . 5 ⊢ (𝑦 = 𝑥 → (𝜑 ↔ 𝜓)) | |
3 | 2 | ralab 3687 | . . . 4 ⊢ (∀𝑥 ∈ {𝑦 ∣ 𝜑} ¬ 𝜒 ↔ ∀𝑥(𝜓 → ¬ 𝜒)) |
4 | 1, 3 | xchbinx 333 | . . 3 ⊢ (∃𝑥 ∈ {𝑦 ∣ 𝜑}𝜒 ↔ ¬ ∀𝑥(𝜓 → ¬ 𝜒)) |
5 | imnang 1844 | . . 3 ⊢ (∀𝑥(𝜓 → ¬ 𝜒) ↔ ∀𝑥 ¬ (𝜓 ∧ 𝜒)) | |
6 | 4, 5 | xchbinx 333 | . 2 ⊢ (∃𝑥 ∈ {𝑦 ∣ 𝜑}𝜒 ↔ ¬ ∀𝑥 ¬ (𝜓 ∧ 𝜒)) |
7 | df-ex 1782 | . 2 ⊢ (∃𝑥(𝜓 ∧ 𝜒) ↔ ¬ ∀𝑥 ¬ (𝜓 ∧ 𝜒)) | |
8 | 6, 7 | bitr4i 277 | 1 ⊢ (∃𝑥 ∈ {𝑦 ∣ 𝜑}𝜒 ↔ ∃𝑥(𝜓 ∧ 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∀wal 1539 ∃wex 1781 {cab 2709 ∀wral 3061 ∃wrex 3070 |
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 1913 ax-6 1971 ax-7 2011 |
This theorem depends on definitions: df-bi 206 df-an 397 df-ex 1782 df-sb 2068 df-clab 2710 df-ral 3062 df-rex 3071 |
This theorem is referenced by: 4sqlem12 16893 noinfno 27445 sleadd1 27699 addsuniflem 27711 addsasslem1 27713 addsasslem2 27714 mulsuniflem 27831 addsdilem1 27833 addsdilem2 27834 mulsasslem1 27845 mulsasslem2 27846 mblfinlem3 36830 mblfinlem4 36831 ismblfin 36832 itg2addnclem 36842 itg2addnc 36845 diophrex 41815 |
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