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Theorem bj-ax12v3 32800
Description: A weak version of ax-12 2087 which is stronger than ax12v 2088. Note that if one assumes reflexivity of equality 𝑥 = 𝑥 (equid 1985), then bj-ax12v3 32800 implies ax-5 1879 over modal logic K (substitute 𝑥 for 𝑦). See also bj-ax12v3ALT 32801. (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 1879 . 2 (𝜑 → ∀𝑦𝜑)
2 ax12 2340 . 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 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  ax-10 2059  ax-12 2087  ax-13 2282
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ex 1745  df-nf 1750
This theorem is referenced by: (None)
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