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Mirrors > Home > ILE Home > Th. List > rexbii | GIF version |
Description: Inference adding restricted existential quantifier to both sides of an equivalence. (Contributed by NM, 23-Nov-1994.) (Revised by Mario Carneiro, 17-Oct-2016.) |
Ref | Expression |
---|---|
ralbii.1 | ⊢ (𝜑 ↔ 𝜓) |
Ref | Expression |
---|---|
rexbii | ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐴 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ralbii.1 | . . . 4 ⊢ (𝜑 ↔ 𝜓) | |
2 | 1 | a1i 9 | . . 3 ⊢ (⊤ → (𝜑 ↔ 𝜓)) |
3 | 2 | rexbidv 2458 | . 2 ⊢ (⊤ → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐴 𝜓)) |
4 | 3 | mptru 1344 | 1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐴 𝜓) |
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