Theorem bj-ax12v3 37108

Description: A weak version of ax-12 2206 which is stronger than ax12v 2207. Note that if one assumes reflexivity of equality ⊢ 𝑥 = 𝑥 (equid 2026), then bj-ax12v3 37108 implies ax-5 1924 over modal logic K (substitute 𝑥 for 𝑦). See also bj-ax12v3ALT 37109. (Contributed by BJ, 6-Jul-2021.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-ax12v3	⊢ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))

Distinct variable group:   𝜑,𝑦

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem bj-ax12v3

Step	Hyp	Ref	Expression
1		ax-5 1924	. 2 ⊢ (𝜑 → ∀𝑦𝜑)
2		ax12 2448	. 2 ⊢ (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
3	1, 2	syl5 34	1 ⊢ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))

Syntax hints:   → wi 4  ∀wal 1552

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-10 2169  ax-12 2206  ax-13 2397

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1794  df-nf 1798

This theorem is referenced by: (None)