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Theorem spcdvw 42224
 Description: A version of spcdv 3264 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 . . . . 5 (𝑥 = 𝐴 → (𝜓𝜒))
21biimpd 218 . . . 4 (𝑥 = 𝐴 → (𝜓𝜒))
32ax-gen 1713 . . 3 𝑥(𝑥 = 𝐴 → (𝜓𝜒))
43a1i 11 . 2 (𝜑 → ∀𝑥(𝑥 = 𝐴 → (𝜓𝜒)))
5 spcdvw.1 . 2 (𝜑𝐴𝐵)
6 nfv 1830 . . 3 𝑥𝜒
7 nfcv 2751 . . 3 𝑥𝐴
86, 7spcimgft 3257 . 2 (∀𝑥(𝑥 = 𝐴 → (𝜓𝜒)) → (𝐴𝐵 → (∀𝑥𝜓𝜒)))
94, 5, 8sylc 63 1 (𝜑 → (∀𝑥𝜓𝜒))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195  ∀wal 1473   = wceq 1475   ∈ wcel 1977 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175 This theorem is referenced by:  setrec1lem4  42236
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