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Theorem bj-ssbjustlem 33153
 Description: Lemma for bj-ssbjust 33154. (Contributed by BJ, 9-Nov-2021.)
Assertion
Ref Expression
bj-ssbjustlem (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)) ↔ ∀𝑧(𝑧 = 𝑡 → ∀𝑥(𝑥 = 𝑧𝜑)))
Distinct variable groups:   𝑥,𝑦,𝑧   𝑦,𝑡,𝑧   𝜑,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑡)

Proof of Theorem bj-ssbjustlem
StepHypRef Expression
1 equequ1 2129 . . 3 (𝑦 = 𝑧 → (𝑦 = 𝑡𝑧 = 𝑡))
2 equequ2 2130 . . . . 5 (𝑦 = 𝑧 → (𝑥 = 𝑦𝑥 = 𝑧))
32imbi1d 333 . . . 4 (𝑦 = 𝑧 → ((𝑥 = 𝑦𝜑) ↔ (𝑥 = 𝑧𝜑)))
43albidv 2019 . . 3 (𝑦 = 𝑧 → (∀𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑧𝜑)))
51, 4imbi12d 336 . 2 (𝑦 = 𝑧 → ((𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)) ↔ (𝑧 = 𝑡 → ∀𝑥(𝑥 = 𝑧𝜑))))
65cbvalvw 2143 1 (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)) ↔ ∀𝑧(𝑧 = 𝑡 → ∀𝑥(𝑥 = 𝑧𝜑)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198  ∀wal 1654 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112 This theorem depends on definitions:  df-bi 199  df-an 387  df-ex 1879 This theorem is referenced by:  bj-ssbjust  33154
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