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Theorem spcdvw 42936
Description: A version of spcdv 3431 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 219 . . 3 (𝑥 = 𝐴 → (𝜓𝜒))
32ax-gen 1871 . 2 𝑥(𝑥 = 𝐴 → (𝜓𝜒))
4 spcdvw.1 . 2 (𝜑𝐴𝐵)
5 nfv 1992 . . 3 𝑥𝜒
6 nfcv 2902 . . 3 𝑥𝐴
75, 6spcimgft 3424 . 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 196  wal 1630   = wceq 1632  wcel 2139
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-v 3342
This theorem is referenced by:  setrec1lem4  42947
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