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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 3472 | . 2 ⊢ (∃𝑥 ∈ V 𝜑 ↔ ∃𝑦 ∈ V [𝑦 / 𝑥]𝜑) | |
2 | rexv 3522 | . 2 ⊢ (∃𝑥 ∈ V 𝜑 ↔ ∃𝑥𝜑) | |
3 | rexv 3522 | . 2 ⊢ (∃𝑦 ∈ V [𝑦 / 𝑥]𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑) | |
4 | 1, 2, 3 | 3bitr3i 303 | 1 ⊢ (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∃wex 1780 [wsb 2069 ∃wrex 3141 Vcvv 3496 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-13 2390 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-v 3498 |
This theorem is referenced by: onfrALTlem1 40889 onfrALTlem1VD 41231 |
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