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Mirrors > Home > MPE Home > Th. List > rmoanid | Structured version Visualization version GIF version |
Description: Cancellation law for restricted at-most-one quantification. (Contributed by Peter Mazsa, 24-May-2018.) |
Ref | Expression |
---|---|
rmoanid | ⊢ (∃*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 | mobii 2631 | . 2 ⊢ (∃*𝑥(𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝜑)) ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) |
3 | df-rmo 3148 | . 2 ⊢ (∃*𝑥 ∈ 𝐴 (𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝜑))) | |
4 | df-rmo 3148 | . 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 2114 ∃*wmo 2620 ∃*wrmo 3143 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1781 df-mo 2622 df-rmo 3148 |
This theorem is referenced by: (None) |
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