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Theorem cbvrexsvwOLD 3308
Description: Obsolete version of cbvrexsvw 3306 as of 8-Mar-2025. (Contributed by NM, 20-Nov-2005.) Avoid ax-13 2365. (Revised by Gino Giotto, 10-Jan-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
cbvrexsvwOLD (∃𝑥𝐴 𝜑 ↔ ∃𝑦𝐴 [𝑦 / 𝑥]𝜑)
Distinct variable groups:   𝑥,𝐴,𝑦   𝜑,𝑦   𝑥,𝑦
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem cbvrexsvwOLD
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 nfv 1909 . . 3 𝑧𝜑
2 nfs1v 2145 . . 3 𝑥[𝑧 / 𝑥]𝜑
3 sbequ12 2238 . . 3 (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
41, 2, 3cbvrexw 3295 . 2 (∃𝑥𝐴 𝜑 ↔ ∃𝑧𝐴 [𝑧 / 𝑥]𝜑)
5 nfv 1909 . . 3 𝑦[𝑧 / 𝑥]𝜑
6 nfv 1909 . . 3 𝑧[𝑦 / 𝑥]𝜑
7 sbequ 2078 . . 3 (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
85, 6, 7cbvrexw 3295 . 2 (∃𝑧𝐴 [𝑧 / 𝑥]𝜑 ↔ ∃𝑦𝐴 [𝑦 / 𝑥]𝜑)
94, 8bitri 274 1 (∃𝑥𝐴 𝜑 ↔ ∃𝑦𝐴 [𝑦 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 205  [wsb 2059  wrex 3060
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-10 2129  ax-11 2146  ax-12 2166
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-ex 1774  df-nf 1778  df-sb 2060  df-clel 2802  df-nfc 2877  df-ral 3052  df-rex 3061
This theorem is referenced by: (None)
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