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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 |
---|---|
raleq1d.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
rexeqdv | ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | raleq1d.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | rexeq 3310 | . 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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