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Mirrors > Home > MPE Home > Th. List > reuanid | Structured version Visualization version GIF version |
Description: Cancellation law for restricted unique existential quantification. (Contributed by Peter Mazsa, 12-Feb-2018.) |
Ref | Expression |
---|---|
reuanid | ⊢ (∃!𝑥 ∈ 𝐴 (𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃!𝑥 ∈ 𝐴 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | anabs5 661 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝜑)) ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)) | |
2 | 1 | eubii 2669 | . 2 ⊢ (∃!𝑥(𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝜑)) ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) |
3 | df-reu 3144 | . 2 ⊢ (∃!𝑥 ∈ 𝐴 (𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝜑))) | |
4 | df-reu 3144 | . 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
5 | 2, 3, 4 | 3bitr4i 305 | 1 ⊢ (∃!𝑥 ∈ 𝐴 (𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃!𝑥 ∈ 𝐴 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 ∈ wcel 2113 ∃!weu 2652 ∃!wreu 3139 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1780 df-mo 2621 df-eu 2653 df-reu 3144 |
This theorem is referenced by: (None) |
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