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Mirrors > Home > MPE Home > Th. List > reueqd | Structured version Visualization version GIF version |
Description: Equality deduction for restricted unique existential quantifier. (Contributed by NM, 5-Apr-2004.) |
Ref | Expression |
---|---|
raleqd.1 | ⊢ (𝐴 = 𝐵 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
reueqd | ⊢ (𝐴 = 𝐵 → (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥 ∈ 𝐵 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reueq1 3406 | . 2 ⊢ (𝐴 = 𝐵 → (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥 ∈ 𝐵 𝜑)) | |
2 | raleqd.1 | . . 3 ⊢ (𝐴 = 𝐵 → (𝜑 ↔ 𝜓)) | |
3 | 2 | reubidv 3388 | . 2 ⊢ (𝐴 = 𝐵 → (∃!𝑥 ∈ 𝐵 𝜑 ↔ ∃!𝑥 ∈ 𝐵 𝜓)) |
4 | 1, 3 | bitrd 281 | 1 ⊢ (𝐴 = 𝐵 → (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥 ∈ 𝐵 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1531 ∃!wreu 3138 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-ext 2791 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1775 df-mo 2616 df-eu 2648 df-cleq 2812 df-clel 2891 df-reu 3143 |
This theorem is referenced by: aceq1 9535 |
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