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Theorem bj-ssblem2 32756
Description: An instance of ax-11 2074 proved without it. The converse may not be provable without ax-11 2074 (since using alcomiw 2013 would require a DV on 𝜑, 𝑥, which defeats the purpose). (Contributed by BJ, 22-Dec-2020.)
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 1998 . . 3 (𝑦 = 𝑧 → (𝑦 = 𝑡𝑧 = 𝑡))
2 equequ2 1999 . . . 4 (𝑦 = 𝑧 → (𝑥 = 𝑦𝑥 = 𝑧))
32imbi1d 330 . . 3 (𝑦 = 𝑧 → ((𝑥 = 𝑦𝜑) ↔ (𝑥 = 𝑧𝜑)))
41, 3imbi12d 333 . 2 (𝑦 = 𝑧 → ((𝑦 = 𝑡 → (𝑥 = 𝑦𝜑)) ↔ (𝑧 = 𝑡 → (𝑥 = 𝑧𝜑))))
54alcomiw 2013 1 (∀𝑥𝑦(𝑦 = 𝑡 → (𝑥 = 𝑦𝜑)) → ∀𝑦𝑥(𝑦 = 𝑡 → (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:  bj-ssb1a  32757
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