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Theorem sbcimdv 2817
Description: Substitution analog of Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 11-Nov-2005.)
Hypothesis
Ref Expression
sbcimdv.1 (φ → (ψχ))
Assertion
Ref Expression
sbcimdv ((φ A 𝑉) → ([A / x]ψ[A / x]χ))
Distinct variable group:   φ,x
Allowed substitution hints:   ψ(x)   χ(x)   A(x)   𝑉(x)

Proof of Theorem sbcimdv
StepHypRef Expression
1 sbcimdv.1 . . . . 5 (φ → (ψχ))
21alrimiv 1751 . . . 4 (φx(ψχ))
3 spsbc 2769 . . . 4 (A 𝑉 → (x(ψχ) → [A / x](ψχ)))
42, 3syl5 28 . . 3 (A 𝑉 → (φ[A / x](ψχ)))
5 sbcimg 2798 . . 3 (A 𝑉 → ([A / x](ψχ) ↔ ([A / x]ψ[A / x]χ)))
64, 5sylibd 138 . 2 (A 𝑉 → (φ → ([A / x]ψ[A / x]χ)))
76impcom 116 1 ((φ A 𝑉) → ([A / x]ψ[A / x]χ))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wal 1240   wcel 1390  [wsbc 2758
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-sbc 2759
This theorem is referenced by: (None)
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