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Theorem wl-dfralflem 35043
 Description: Lemma for wl-dfralf 35044 and wl-dfralv . (Contributed by Wolf Lammen, 23-May-2023.)
Assertion
Ref Expression
wl-dfralflem (∀𝑦𝑥(𝑦𝐴 → (𝑥 = 𝑦𝜑)) ↔ ∀𝑥(𝑥𝐴𝜑))
Distinct variable groups:   𝑥,𝑦   𝜑,𝑦   𝑦,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)

Proof of Theorem wl-dfralflem
StepHypRef Expression
1 alcom 2160 . 2 (∀𝑦𝑥(𝑦𝐴 → (𝑥 = 𝑦𝜑)) ↔ ∀𝑥𝑦(𝑦𝐴 → (𝑥 = 𝑦𝜑)))
2 bi2.04 392 . . . . . 6 ((𝑦𝐴 → (𝑥 = 𝑦𝜑)) ↔ (𝑥 = 𝑦 → (𝑦𝐴𝜑)))
3 equcom 2025 . . . . . . 7 (𝑥 = 𝑦𝑦 = 𝑥)
43imbi1i 353 . . . . . 6 ((𝑥 = 𝑦 → (𝑦𝐴𝜑)) ↔ (𝑦 = 𝑥 → (𝑦𝐴𝜑)))
52, 4bitri 278 . . . . 5 ((𝑦𝐴 → (𝑥 = 𝑦𝜑)) ↔ (𝑦 = 𝑥 → (𝑦𝐴𝜑)))
65albii 1821 . . . 4 (∀𝑦(𝑦𝐴 → (𝑥 = 𝑦𝜑)) ↔ ∀𝑦(𝑦 = 𝑥 → (𝑦𝐴𝜑)))
7 eleq1w 2872 . . . . . 6 (𝑦 = 𝑥 → (𝑦𝐴𝑥𝐴))
87imbi1d 345 . . . . 5 (𝑦 = 𝑥 → ((𝑦𝐴𝜑) ↔ (𝑥𝐴𝜑)))
98equsalvw 2010 . . . 4 (∀𝑦(𝑦 = 𝑥 → (𝑦𝐴𝜑)) ↔ (𝑥𝐴𝜑))
106, 9bitri 278 . . 3 (∀𝑦(𝑦𝐴 → (𝑥 = 𝑦𝜑)) ↔ (𝑥𝐴𝜑))
1110albii 1821 . 2 (∀𝑥𝑦(𝑦𝐴 → (𝑥 = 𝑦𝜑)) ↔ ∀𝑥(𝑥𝐴𝜑))
121, 11bitri 278 1 (∀𝑦𝑥(𝑦𝐴 → (𝑥 = 𝑦𝜑)) ↔ ∀𝑥(𝑥𝐴𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1536   ∈ wcel 2111 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-11 2158 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-clel 2870 This theorem is referenced by:  wl-dfralf  35044  wl-dfralv  35046
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