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Mirrors > Home > ILE Home > Th. List > rmoeqd | GIF version |
Description: Equality deduction for restricted at-most-one quantifier. (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
Ref | Expression |
---|---|
raleqd.1 | ⊢ (𝐴 = 𝐵 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
rmoeqd | ⊢ (𝐴 = 𝐵 → (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥 ∈ 𝐵 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rmoeq1 2675 | . 2 ⊢ (𝐴 = 𝐵 → (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥 ∈ 𝐵 𝜑)) | |
2 | raleqd.1 | . . 3 ⊢ (𝐴 = 𝐵 → (𝜑 ↔ 𝜓)) | |
3 | 2 | rmobidv 2665 | . 2 ⊢ (𝐴 = 𝐵 → (∃*𝑥 ∈ 𝐵 𝜑 ↔ ∃*𝑥 ∈ 𝐵 𝜓)) |
4 | 1, 3 | bitrd 188 | 1 ⊢ (𝐴 = 𝐵 → (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥 ∈ 𝐵 𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 = wceq 1353 ∃*wrmo 2458 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-cleq 2170 df-clel 2173 df-nfc 2308 df-rmo 2463 |
This theorem is referenced by: (None) |
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