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Theorem sbjust 2069
Description: Justification theorem for df-sb 2071 proved from Tarski's FOL axiom schemes. (Contributed by BJ, 22-Jan-2023.)
Assertion
Ref Expression
sbjust (∀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 sbjust
StepHypRef Expression
1 equequ1 2033 . . 3 (𝑦 = 𝑧 → (𝑦 = 𝑡𝑧 = 𝑡))
2 equequ2 2034 . . . . 5 (𝑦 = 𝑧 → (𝑥 = 𝑦𝑥 = 𝑧))
32imbi1d 345 . . . 4 (𝑦 = 𝑧 → ((𝑥 = 𝑦𝜑) ↔ (𝑥 = 𝑧𝜑)))
43albidv 1928 . . 3 (𝑦 = 𝑧 → (∀𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑧𝜑)))
51, 4imbi12d 348 . 2 (𝑦 = 𝑧 → ((𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)) ↔ (𝑧 = 𝑡 → ∀𝑥(𝑥 = 𝑧𝜑))))
65cbvalvw 2044 1 (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)) ↔ ∀𝑧(𝑧 = 𝑡 → ∀𝑥(𝑥 = 𝑧𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wal 1541
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1788
This theorem is referenced by: (None)
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