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Theorem axc11rvOLD 2305
Description: Obsolete proof of axc11rv 2304 as of 11-Oct-2021. (Contributed by BJ, 6-Jul-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
axc11rvOLD (∀𝑥 𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥𝜑))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem axc11rvOLD
StepHypRef Expression
1 ax12v2 2205 . . . 4 (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
21sps 2209 . . 3 (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
3 pm2.27 42 . . . 4 (𝑥 = 𝑦 → ((𝑥 = 𝑦𝜑) → 𝜑))
43al2imi 1891 . . 3 (∀𝑥 𝑥 = 𝑦 → (∀𝑥(𝑥 = 𝑦𝜑) → ∀𝑥𝜑))
52, 4syld 47 . 2 (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))
65spsd 2211 1 (∀𝑥 𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥𝜑))
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-12 2203
This theorem depends on definitions:  df-bi 197  df-an 383  df-ex 1853
This theorem is referenced by:  axc16gOLD  2324
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