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| Mirrors > Home > MPE Home > Th. List > rexeqdv | Structured version Visualization version GIF version | ||
| Description: Equality deduction for restricted existential quantifier. (Contributed by NM, 14-Jan-2007.) |
| Ref | Expression |
|---|---|
| raleqdv.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
| Ref | Expression |
|---|---|
| rexeqdv | ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜓)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | raleqdv.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
| 2 | rexeq 3322 | . 2 ⊢ (𝐴 = 𝐵 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜓)) | |
| 3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜓)) |
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