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Theorem bj-ax12 34104
Description: Remove a DV condition from bj-ax12v 34103 (using core axioms only). (Contributed by BJ, 26-Dec-2020.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-ax12 𝑥(𝑥 = 𝑡 → (𝜑 → ∀𝑥(𝑥 = 𝑡𝜑)))
Distinct variable group:   𝑥,𝑡
Allowed substitution hints:   𝜑(𝑥,𝑡)

Proof of Theorem bj-ax12
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 bj-ax12v 34103 . . 3 𝑥(𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
2 equequ2 2033 . . . . 5 (𝑦 = 𝑡 → (𝑥 = 𝑦𝑥 = 𝑡))
32imbi1d 345 . . . . . . 7 (𝑦 = 𝑡 → ((𝑥 = 𝑦𝜑) ↔ (𝑥 = 𝑡𝜑)))
43albidv 1921 . . . . . 6 (𝑦 = 𝑡 → (∀𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑡𝜑)))
54imbi2d 344 . . . . 5 (𝑦 = 𝑡 → ((𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)) ↔ (𝜑 → ∀𝑥(𝑥 = 𝑡𝜑))))
62, 5imbi12d 348 . . . 4 (𝑦 = 𝑡 → ((𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))) ↔ (𝑥 = 𝑡 → (𝜑 → ∀𝑥(𝑥 = 𝑡𝜑)))))
76albidv 1921 . . 3 (𝑦 = 𝑡 → (∀𝑥(𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))) ↔ ∀𝑥(𝑥 = 𝑡 → (𝜑 → ∀𝑥(𝑥 = 𝑡𝜑)))))
81, 7mpbii 236 . 2 (𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑡 → (𝜑 → ∀𝑥(𝑥 = 𝑡𝜑))))
9 ax6ev 1972 . 2 𝑦 𝑦 = 𝑡
108, 9exlimiiv 1932 1 𝑥(𝑥 = 𝑡 → (𝜑 → ∀𝑥(𝑥 = 𝑡𝜑)))
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 1911  ax-6 1970  ax-7 2015  ax-12 2176
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782
This theorem is referenced by:  bj-ax12ssb  34105  bj-sb56  34108
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