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Mirrors > Home > NFE Home > Th. List > cbvrmo | 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 2832 | . . 3 ⊢ (∃x ∈ A φ ↔ ∃y ∈ A ψ) |
5 | 1, 2, 3 | cbvreu 2833 | . . 3 ⊢ (∃!x ∈ A φ ↔ ∃!y ∈ A ψ) |
6 | 4, 5 | imbi12i 316 | . 2 ⊢ ((∃x ∈ A φ → ∃!x ∈ A φ) ↔ (∃y ∈ A ψ → ∃!y ∈ A ψ)) |
7 | rmo5 2827 | . 2 ⊢ (∃*x ∈ A φ ↔ (∃x ∈ A φ → ∃!x ∈ A φ)) | |
8 | rmo5 2827 | . 2 ⊢ (∃*y ∈ A ψ ↔ (∃y ∈ A ψ → ∃!y ∈ A ψ)) | |
9 | 6, 7, 8 | 3bitr4i 268 | 1 ⊢ (∃*x ∈ A φ ↔ ∃*y ∈ A ψ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 Ⅎwnf 1544 ∃wrex 2615 ∃!wreu 2616 ∃*wrmo 2617 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-eu 2208 df-mo 2209 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ral 2619 df-rex 2620 df-reu 2621 df-rmo 2622 |
This theorem is referenced by: cbvrmov 2838 |
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