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Mathbox for Alan Sare |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cbvexsv | Structured version Visualization version GIF version |
Description: A theorem pertaining to the substitution for an existentially quantified variable when the substituted variable does not occur in the quantified wff. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
cbvexsv | ⊢ (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbvrexsv 3364 | . 2 ⊢ (∃𝑥 ∈ V 𝜑 ↔ ∃𝑦 ∈ V [𝑦 / 𝑥]𝜑) | |
2 | rexv 3506 | . 2 ⊢ (∃𝑥 ∈ V 𝜑 ↔ ∃𝑥𝜑) | |
3 | rexv 3506 | . 2 ⊢ (∃𝑦 ∈ V [𝑦 / 𝑥]𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑) | |
4 | 1, 2, 3 | 3bitr3i 301 | 1 ⊢ (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∃wex 1775 [wsb 2061 ∃wrex 3067 Vcvv 3477 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-13 2374 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1539 df-ex 1776 df-nf 1780 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ral 3059 df-rex 3068 df-v 3479 |
This theorem is referenced by: onfrALTlem1 44545 onfrALTlem1VD 44887 |
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