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Theorem bj-ssblem2 34015
 Description: An instance of ax-11 2162 proved without it. The converse may not be provable without ax-11 2162 (since using alcomiw 2051 would require a DV on 𝜑, 𝑥, which defeats the purpose). (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-ssblem2 (∀𝑥𝑦(𝑦 = 𝑡 → (𝑥 = 𝑦𝜑)) → ∀𝑦𝑥(𝑦 = 𝑡 → (𝑥 = 𝑦𝜑)))
Distinct variable groups:   𝑥,𝑦,𝑡   𝜑,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑡)

Proof of Theorem bj-ssblem2
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 equequ1 2033 . . 3 (𝑦 = 𝑧 → (𝑦 = 𝑡𝑧 = 𝑡))
2 equequ2 2034 . . . 4 (𝑦 = 𝑧 → (𝑥 = 𝑦𝑥 = 𝑧))
32imbi1d 345 . . 3 (𝑦 = 𝑧 → ((𝑥 = 𝑦𝜑) ↔ (𝑥 = 𝑧𝜑)))
41, 3imbi12d 348 . 2 (𝑦 = 𝑧 → ((𝑦 = 𝑡 → (𝑥 = 𝑦𝜑)) ↔ (𝑧 = 𝑡 → (𝑥 = 𝑧𝜑))))
54alcomiw 2051 1 (∀𝑥𝑦(𝑦 = 𝑡 → (𝑥 = 𝑦𝜑)) → ∀𝑦𝑥(𝑦 = 𝑡 → (𝑥 = 𝑦𝜑)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1536 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 1912  ax-6 1971  ax-7 2016 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782 This theorem is referenced by: (None)
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