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Theorem stoweidlem4 45454
Description: Lemma for stoweid 45513: 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 2813 . . . . 5 (𝑥 = 𝐵 → (𝑥 ∈ ℝ ↔ 𝐵 ∈ ℝ))
21anbi2d 628 . . . 4 (𝑥 = 𝐵 → ((𝜑𝑥 ∈ ℝ) ↔ (𝜑𝐵 ∈ ℝ)))
3 simpl 481 . . . . . 6 ((𝑥 = 𝐵𝑡𝑇) → 𝑥 = 𝐵)
43mpteq2dva 5243 . . . . 5 (𝑥 = 𝐵 → (𝑡𝑇𝑥) = (𝑡𝑇𝐵))
54eleq1d 2810 . . . 4 (𝑥 = 𝐵 → ((𝑡𝑇𝑥) ∈ 𝐴 ↔ (𝑡𝑇𝐵) ∈ 𝐴))
62, 5imbi12d 343 . . 3 (𝑥 = 𝐵 → (((𝜑𝑥 ∈ ℝ) → (𝑡𝑇𝑥) ∈ 𝐴) ↔ ((𝜑𝐵 ∈ ℝ) → (𝑡𝑇𝐵) ∈ 𝐴)))
7 stoweidlem4.1 . . 3 ((𝜑𝑥 ∈ ℝ) → (𝑡𝑇𝑥) ∈ 𝐴)
86, 7vtoclg 3533 . 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 394   = wceq 1533  wcel 2098  cmpt 5226  cr 11135
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-opab 5206  df-mpt 5227
This theorem is referenced by:  stoweidlem18  45468  stoweidlem19  45469  stoweidlem22  45472  stoweidlem32  45482  stoweidlem36  45486  stoweidlem40  45490  stoweidlem41  45491  stoweidlem55  45505
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