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Theorem spcdvw 47979
Description: A version of spcdv 3578 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 228 . . 3 (𝑥 = 𝐴 → (𝜓𝜒))
32ax-gen 1789 . 2 𝑥(𝑥 = 𝐴 → (𝜓𝜒))
4 spcdvw.1 . 2 (𝜑𝐴𝐵)
5 nfv 1909 . . 3 𝑥𝜒
6 nfcv 2897 . . 3 𝑥𝐴
75, 6spcimgft 3571 . 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 205  wal 1531   = wceq 1533  wcel 2098
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-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-v 3470
This theorem is referenced by:  setrec1lem4  47990
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