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Theorem bj-ssblem1 32307
Description: A lemma for the definiens of df-sb 1878. An instance of sp 2051 proved without it. Note: it has a common subproof with bj-ssbjust 32295. (Contributed by BJ, 22-Dec-2020.)
Assertion
Ref Expression
bj-ssblem1 (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)) → (𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)))
Distinct variable groups:   𝑥,𝑦   𝑦,𝑡   𝜑,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑡)

Proof of Theorem bj-ssblem1
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 equequ1 1949 . . 3 (𝑦 = 𝑧 → (𝑦 = 𝑡𝑧 = 𝑡))
2 equequ2 1950 . . . . 5 (𝑦 = 𝑧 → (𝑥 = 𝑦𝑥 = 𝑧))
32imbi1d 331 . . . 4 (𝑦 = 𝑧 → ((𝑥 = 𝑦𝜑) ↔ (𝑥 = 𝑧𝜑)))
43albidv 1846 . . 3 (𝑦 = 𝑧 → (∀𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑧𝜑)))
51, 4imbi12d 334 . 2 (𝑦 = 𝑧 → ((𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)) ↔ (𝑧 = 𝑡 → ∀𝑥(𝑥 = 𝑧𝜑))))
65spw 1964 1 (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)) → (𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1478
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 1836  ax-6 1885  ax-7 1932
This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1702
This theorem is referenced by: (None)
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