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Theorem bj-ax12v3 34435
 Description: A weak version of ax-12 2175 which is stronger than ax12v 2176. Note that if one assumes reflexivity of equality ⊢ 𝑥 = 𝑥 (equid 2019), then bj-ax12v3 34435 implies ax-5 1911 over modal logic K (substitute 𝑥 for 𝑦). See also bj-ax12v3ALT 34436. (Contributed by BJ, 6-Jul-2021.) (Proof modification is discouraged.)
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 1911 . 2 (𝜑 → ∀𝑦𝜑)
2 ax12 2434 . 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 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-10 2142  ax-12 2175  ax-13 2379 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-nf 1786 This theorem is referenced by: (None)
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