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Theorem stoweidlem4 39515
Description: Lemma for stoweid 39574: 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 2692 . . . . 5 (𝑥 = 𝐵 → (𝑥 ∈ ℝ ↔ 𝐵 ∈ ℝ))
21anbi2d 739 . . . 4 (𝑥 = 𝐵 → ((𝜑𝑥 ∈ ℝ) ↔ (𝜑𝐵 ∈ ℝ)))
3 simpl 473 . . . . . 6 ((𝑥 = 𝐵𝑡𝑇) → 𝑥 = 𝐵)
43mpteq2dva 4709 . . . . 5 (𝑥 = 𝐵 → (𝑡𝑇𝑥) = (𝑡𝑇𝐵))
54eleq1d 2688 . . . 4 (𝑥 = 𝐵 → ((𝑡𝑇𝑥) ∈ 𝐴 ↔ (𝑡𝑇𝐵) ∈ 𝐴))
62, 5imbi12d 334 . . 3 (𝑥 = 𝐵 → (((𝜑𝑥 ∈ ℝ) → (𝑡𝑇𝑥) ∈ 𝐴) ↔ ((𝜑𝐵 ∈ ℝ) → (𝑡𝑇𝐵) ∈ 𝐴)))
7 stoweidlem4.1 . . 3 ((𝜑𝑥 ∈ ℝ) → (𝑡𝑇𝑥) ∈ 𝐴)
86, 7vtoclg 3257 . 2 (𝐵 ∈ ℝ → ((𝜑𝐵 ∈ ℝ) → (𝑡𝑇𝐵) ∈ 𝐴))
98anabsi7 859 1 ((𝜑𝐵 ∈ ℝ) → (𝑡𝑇𝐵) ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1992  cmpt 4678  cr 9880
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-clab 2613  df-cleq 2619  df-clel 2622  df-ral 2917  df-v 3193  df-opab 4679  df-mpt 4680
This theorem is referenced by:  stoweidlem18  39529  stoweidlem19  39530  stoweidlem22  39533  stoweidlem32  39543  stoweidlem36  39547  stoweidlem40  39551  stoweidlem41  39552  stoweidlem55  39566
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