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Theorem stoweidlem4 40641
Description: Lemma for stoweid 40700: a class variable replaces a setvar variable, for constant functions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
Hypothesis
Ref Expression
stoweidlem4.1 ((𝜑𝑥 ∈ ℝ) → (𝑡𝑇𝑥) ∈ 𝐴)
Assertion
Ref Expression
stoweidlem4 ((𝜑𝐵 ∈ ℝ) → (𝑡𝑇𝐵) ∈ 𝐴)
Distinct variable groups:   𝑥,𝑡,𝐵   𝑥,𝐴   𝑥,𝑇   𝜑,𝑥
Allowed substitution hints:   𝜑(𝑡)   𝐴(𝑡)   𝑇(𝑡)

Proof of Theorem stoweidlem4
StepHypRef Expression
1 eleq1 2791 . . . . 5 (𝑥 = 𝐵 → (𝑥 ∈ ℝ ↔ 𝐵 ∈ ℝ))
21anbi2d 742 . . . 4 (𝑥 = 𝐵 → ((𝜑𝑥 ∈ ℝ) ↔ (𝜑𝐵 ∈ ℝ)))
3 simpl 474 . . . . . 6 ((𝑥 = 𝐵𝑡𝑇) → 𝑥 = 𝐵)
43mpteq2dva 4852 . . . . 5 (𝑥 = 𝐵 → (𝑡𝑇𝑥) = (𝑡𝑇𝐵))
54eleq1d 2788 . . . 4 (𝑥 = 𝐵 → ((𝑡𝑇𝑥) ∈ 𝐴 ↔ (𝑡𝑇𝐵) ∈ 𝐴))
62, 5imbi12d 333 . . 3 (𝑥 = 𝐵 → (((𝜑𝑥 ∈ ℝ) → (𝑡𝑇𝑥) ∈ 𝐴) ↔ ((𝜑𝐵 ∈ ℝ) → (𝑡𝑇𝐵) ∈ 𝐴)))
7 stoweidlem4.1 . . 3 ((𝜑𝑥 ∈ ℝ) → (𝑡𝑇𝑥) ∈ 𝐴)
86, 7vtoclg 3370 . 2 (𝐵 ∈ ℝ → ((𝜑𝐵 ∈ ℝ) → (𝑡𝑇𝐵) ∈ 𝐴))
98anabsi7 895 1 ((𝜑𝐵 ∈ ℝ) → (𝑡𝑇𝐵) ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1596  wcel 2103  cmpt 4837  cr 10048
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-clab 2711  df-cleq 2717  df-clel 2720  df-ral 3019  df-v 3306  df-opab 4821  df-mpt 4838
This theorem is referenced by:  stoweidlem18  40655  stoweidlem19  40656  stoweidlem22  40659  stoweidlem32  40669  stoweidlem36  40673  stoweidlem40  40677  stoweidlem41  40678  stoweidlem55  40692
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