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Mirrors > Home > MPE Home > Th. List > cbvrmo | Structured version Visualization version GIF version |
Description: Change the bound variable of restricted "at most one" using implicit substitution. (Contributed by NM, 16-Jun-2017.) |
Ref | Expression |
---|---|
cbvral.1 | ⊢ Ⅎ𝑦𝜑 |
cbvral.2 | ⊢ Ⅎ𝑥𝜓 |
cbvral.3 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
cbvrmo | ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑦 ∈ 𝐴 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbvral.1 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
2 | cbvral.2 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
3 | cbvral.3 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
4 | 1, 2, 3 | cbvrex 3168 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 𝜓) |
5 | 1, 2, 3 | cbvreu 3169 | . . 3 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑦 ∈ 𝐴 𝜓) |
6 | 4, 5 | imbi12i 340 | . 2 ⊢ ((∃𝑥 ∈ 𝐴 𝜑 → ∃!𝑥 ∈ 𝐴 𝜑) ↔ (∃𝑦 ∈ 𝐴 𝜓 → ∃!𝑦 ∈ 𝐴 𝜓)) |
7 | rmo5 3162 | . 2 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 → ∃!𝑥 ∈ 𝐴 𝜑)) | |
8 | rmo5 3162 | . 2 ⊢ (∃*𝑦 ∈ 𝐴 𝜓 ↔ (∃𝑦 ∈ 𝐴 𝜓 → ∃!𝑦 ∈ 𝐴 𝜓)) | |
9 | 6, 7, 8 | 3bitr4i 292 | 1 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑦 ∈ 𝐴 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 Ⅎwnf 1708 ∃wrex 2913 ∃!wreu 2914 ∃*wrmo 2915 |
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-13 2246 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-sb 1881 df-eu 2474 df-mo 2475 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 |
This theorem is referenced by: cbvrmov 3174 cbvdisj 4630 |
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