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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rexsb | Structured version Visualization version GIF version |
Description: An equivalent expression for restricted existence, analogous to exsb 2353. (Contributed by Alexander van der Vekens, 1-Jul-2017.) |
Ref | Expression |
---|---|
rexsb | ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 ∀𝑥(𝑥 = 𝑦 → 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1915 | . 2 ⊢ Ⅎ𝑦𝜑 | |
2 | nfa1 2146 | . 2 ⊢ Ⅎ𝑥∀𝑥(𝑥 = 𝑦 → 𝜑) | |
3 | ax12v 2170 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) | |
4 | sp 2174 | . . . 4 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → (𝑥 = 𝑦 → 𝜑)) | |
5 | 4 | com12 32 | . . 3 ⊢ (𝑥 = 𝑦 → (∀𝑥(𝑥 = 𝑦 → 𝜑) → 𝜑)) |
6 | 3, 5 | impbid 211 | . 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))) |
7 | 1, 2, 6 | cbvrexw 3302 | 1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 ∀𝑥(𝑥 = 𝑦 → 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∀wal 1537 ∃wrex 3068 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-10 2135 ax-11 2152 ax-12 2169 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-ex 1780 df-nf 1784 df-clel 2808 df-nfc 2883 df-ral 3060 df-rex 3069 |
This theorem is referenced by: rexrsb 46108 2rexsb 46109 |
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