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Theorem spcdvw 43524
 Description: A version of spcdv 3492 where 𝜓 and 𝜒 are direct substitutions of each other. This theorem is useful because it does not require 𝜑 and 𝑥 to be distinct variables. (Contributed by Emmett Weisz, 12-Apr-2020.)
Hypotheses
Ref Expression
spcdvw.1 (𝜑𝐴𝐵)
spcdvw.2 (𝑥 = 𝐴 → (𝜓𝜒))
Assertion
Ref Expression
spcdvw (𝜑 → (∀𝑥𝜓𝜒))
Distinct variable groups:   𝑥,𝐴   𝜒,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)   𝐵(𝑥)

Proof of Theorem spcdvw
StepHypRef Expression
1 spcdvw.2 . . . 4 (𝑥 = 𝐴 → (𝜓𝜒))
21biimpd 221 . . 3 (𝑥 = 𝐴 → (𝜓𝜒))
32ax-gen 1839 . 2 𝑥(𝑥 = 𝐴 → (𝜓𝜒))
4 spcdvw.1 . 2 (𝜑𝐴𝐵)
5 nfv 1957 . . 3 𝑥𝜒
6 nfcv 2933 . . 3 𝑥𝐴
75, 6spcimgft 3485 . 2 (∀𝑥(𝑥 = 𝐴 → (𝜓𝜒)) → (𝐴𝐵 → (∀𝑥𝜓𝜒)))
83, 4, 7mpsyl 68 1 (𝜑 → (∀𝑥𝜓𝜒))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198  ∀wal 1599   = wceq 1601   ∈ wcel 2106 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-ext 2753 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-v 3399 This theorem is referenced by:  setrec1lem4  43535
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