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Mirrors > Home > MPE Home > Th. List > reueq1f | Structured version Visualization version GIF version |
Description: Equality theorem for restricted uniqueness quantifier, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 5-Apr-2004.) (Revised by Andrew Salmon, 11-Jul-2011.) |
Ref | Expression |
---|---|
raleq1f.1 | ⊢ Ⅎ𝑥𝐴 |
raleq1f.2 | ⊢ Ⅎ𝑥𝐵 |
Ref | Expression |
---|---|
reueq1f | ⊢ (𝐴 = 𝐵 → (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥 ∈ 𝐵 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | raleq1f.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
2 | raleq1f.2 | . . . 4 ⊢ Ⅎ𝑥𝐵 | |
3 | 1, 2 | nfeq 2776 | . . 3 ⊢ Ⅎ𝑥 𝐴 = 𝐵 |
4 | eleq2 2690 | . . . 4 ⊢ (𝐴 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) | |
5 | 4 | anbi1d 741 | . . 3 ⊢ (𝐴 = 𝐵 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑥 ∈ 𝐵 ∧ 𝜑))) |
6 | 3, 5 | eubid 2488 | . 2 ⊢ (𝐴 = 𝐵 → (∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃!𝑥(𝑥 ∈ 𝐵 ∧ 𝜑))) |
7 | df-reu 2919 | . 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
8 | df-reu 2919 | . 2 ⊢ (∃!𝑥 ∈ 𝐵 𝜑 ↔ ∃!𝑥(𝑥 ∈ 𝐵 ∧ 𝜑)) | |
9 | 6, 7, 8 | 3bitr4g 303 | 1 ⊢ (𝐴 = 𝐵 → (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥 ∈ 𝐵 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∃!weu 2470 Ⅎwnfc 2751 ∃!wreu 2914 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-eu 2474 df-cleq 2615 df-clel 2618 df-nfc 2753 df-reu 2919 |
This theorem is referenced by: reueq1 3140 |
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