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Theorem stoweidlem4 44710
Description: Lemma for stoweid 44769: 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 2821 . . . . 5 (𝑥 = 𝐵 → (𝑥 ∈ ℝ ↔ 𝐵 ∈ ℝ))
21anbi2d 629 . . . 4 (𝑥 = 𝐵 → ((𝜑𝑥 ∈ ℝ) ↔ (𝜑𝐵 ∈ ℝ)))
3 simpl 483 . . . . . 6 ((𝑥 = 𝐵𝑡𝑇) → 𝑥 = 𝐵)
43mpteq2dva 5248 . . . . 5 (𝑥 = 𝐵 → (𝑡𝑇𝑥) = (𝑡𝑇𝐵))
54eleq1d 2818 . . . 4 (𝑥 = 𝐵 → ((𝑡𝑇𝑥) ∈ 𝐴 ↔ (𝑡𝑇𝐵) ∈ 𝐴))
62, 5imbi12d 344 . . 3 (𝑥 = 𝐵 → (((𝜑𝑥 ∈ ℝ) → (𝑡𝑇𝑥) ∈ 𝐴) ↔ ((𝜑𝐵 ∈ ℝ) → (𝑡𝑇𝐵) ∈ 𝐴)))
7 stoweidlem4.1 . . 3 ((𝜑𝑥 ∈ ℝ) → (𝑡𝑇𝑥) ∈ 𝐴)
86, 7vtoclg 3556 . 2 (𝐵 ∈ ℝ → ((𝜑𝐵 ∈ ℝ) → (𝑡𝑇𝐵) ∈ 𝐴))
98anabsi7 669 1 ((𝜑𝐵 ∈ ℝ) → (𝑡𝑇𝐵) ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  cmpt 5231  cr 11108
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-opab 5211  df-mpt 5232
This theorem is referenced by:  stoweidlem18  44724  stoweidlem19  44725  stoweidlem22  44728  stoweidlem32  44738  stoweidlem36  44742  stoweidlem40  44746  stoweidlem41  44747  stoweidlem55  44761
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