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Theorem stoweidlem4 42644
Description: Lemma for stoweid 42703: 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 2877 . . . . 5 (𝑥 = 𝐵 → (𝑥 ∈ ℝ ↔ 𝐵 ∈ ℝ))
21anbi2d 631 . . . 4 (𝑥 = 𝐵 → ((𝜑𝑥 ∈ ℝ) ↔ (𝜑𝐵 ∈ ℝ)))
3 simpl 486 . . . . . 6 ((𝑥 = 𝐵𝑡𝑇) → 𝑥 = 𝐵)
43mpteq2dva 5125 . . . . 5 (𝑥 = 𝐵 → (𝑡𝑇𝑥) = (𝑡𝑇𝐵))
54eleq1d 2874 . . . 4 (𝑥 = 𝐵 → ((𝑡𝑇𝑥) ∈ 𝐴 ↔ (𝑡𝑇𝐵) ∈ 𝐴))
62, 5imbi12d 348 . . 3 (𝑥 = 𝐵 → (((𝜑𝑥 ∈ ℝ) → (𝑡𝑇𝑥) ∈ 𝐴) ↔ ((𝜑𝐵 ∈ ℝ) → (𝑡𝑇𝐵) ∈ 𝐴)))
7 stoweidlem4.1 . . 3 ((𝜑𝑥 ∈ ℝ) → (𝑡𝑇𝑥) ∈ 𝐴)
86, 7vtoclg 3515 . 2 (𝐵 ∈ ℝ → ((𝜑𝐵 ∈ ℝ) → (𝑡𝑇𝐵) ∈ 𝐴))
98anabsi7 670 1 ((𝜑𝐵 ∈ ℝ) → (𝑡𝑇𝐵) ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2111  cmpt 5110  cr 10525
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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-12 2175  ax-ext 2770
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-ral 3111  df-opab 5093  df-mpt 5111
This theorem is referenced by:  stoweidlem18  42658  stoweidlem19  42659  stoweidlem22  42662  stoweidlem32  42672  stoweidlem36  42676  stoweidlem40  42680  stoweidlem41  42681  stoweidlem55  42695
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