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Theorem axc11rvOLD 2125
 Description: Obsolete proof of axc11rv 2124 as of 11-Oct-2011. (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 2036 . . . 4 (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
21sps 2043 . . 3 (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
3 pm2.27 41 . . . 4 (𝑥 = 𝑦 → ((𝑥 = 𝑦𝜑) → 𝜑))
43al2imi 1733 . . 3 (∀𝑥 𝑥 = 𝑦 → (∀𝑥(𝑥 = 𝑦𝜑) → ∀𝑥𝜑))
52, 4syld 46 . 2 (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))
65spsd 2045 1 (∀𝑥 𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1473 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034 This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696 This theorem is referenced by:  axc16gOLD  2147
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