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Theorem spcdvw 44969
Description: A version of spcdv 3572 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 231 . . 3 (𝑥 = 𝐴 → (𝜓𝜒))
32ax-gen 1796 . 2 𝑥(𝑥 = 𝐴 → (𝜓𝜒))
4 spcdvw.1 . 2 (𝜑𝐴𝐵)
5 nfv 1915 . . 3 𝑥𝜒
6 nfcv 2973 . . 3 𝑥𝐴
75, 6spcimgft 3565 . 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 208  wal 1535   = wceq 1537  wcel 2114
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-ext 2792
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 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-v 3475
This theorem is referenced by:  setrec1lem4  44980
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