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Theorem bj-ax12v3 33012
Description: A weak version of ax-12 2203 which is stronger than ax12v 2204. Note that if one assumes reflexivity of equality 𝑥 = 𝑥 (equid 2097), then bj-ax12v3 33012 implies ax-5 1991 over modal logic K (substitute 𝑥 for 𝑦). See also bj-ax12v3ALT 33013. (Contributed by BJ, 6-Jul-2021.)
Assertion
Ref Expression
bj-ax12v3 (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
Distinct variable group:   𝜑,𝑦
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem bj-ax12v3
StepHypRef Expression
1 ax-5 1991 . 2 (𝜑 → ∀𝑦𝜑)
2 ax12 2460 . 2 (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
31, 2syl5 34 1 (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1629
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-10 2174  ax-12 2203  ax-13 2408
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-ex 1853  df-nf 1858
This theorem is referenced by: (None)
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