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Theorem bj-ssblem1 32755
Description: A lemma for the definiens of df-sb 1938. An instance of sp 2091 proved without it. Note: it has a common subproof with bj-ssbjust 32743. (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 1998 . . 3 (𝑦 = 𝑧 → (𝑦 = 𝑡𝑧 = 𝑡))
2 equequ2 1999 . . . . 5 (𝑦 = 𝑧 → (𝑥 = 𝑦𝑥 = 𝑧))
32imbi1d 330 . . . 4 (𝑦 = 𝑧 → ((𝑥 = 𝑦𝜑) ↔ (𝑥 = 𝑧𝜑)))
43albidv 1889 . . 3 (𝑦 = 𝑧 → (∀𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑧𝜑)))
51, 4imbi12d 333 . 2 (𝑦 = 𝑧 → ((𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)) ↔ (𝑧 = 𝑡 → ∀𝑥(𝑥 = 𝑧𝜑))))
65spw 2009 1 (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)) → (𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1521
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981
This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1745
This theorem is referenced by: (None)
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