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Theorem axc11r 2175
 Description: Same as axc11 2302 but with reversed antecedent. Note the use of ax-12 2034 (and not merely ax12v 2035). (Contributed by NM, 25-Jul-2015.)
Assertion
Ref Expression
axc11r (∀𝑦 𝑦 = 𝑥 → (∀𝑥𝜑 → ∀𝑦𝜑))

Proof of Theorem axc11r
StepHypRef Expression
1 ax-12 2034 . . 3 (𝑦 = 𝑥 → (∀𝑥𝜑 → ∀𝑦(𝑦 = 𝑥𝜑)))
21sps 2043 . 2 (∀𝑦 𝑦 = 𝑥 → (∀𝑥𝜑 → ∀𝑦(𝑦 = 𝑥𝜑)))
3 pm2.27 41 . . 3 (𝑦 = 𝑥 → ((𝑦 = 𝑥𝜑) → 𝜑))
43al2imi 1733 . 2 (∀𝑦 𝑦 = 𝑥 → (∀𝑦(𝑦 = 𝑥𝜑) → ∀𝑦𝜑))
52, 4syld 46 1 (∀𝑦 𝑦 = 𝑥 → (∀𝑥𝜑 → ∀𝑦𝜑))
 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-ex 1696 This theorem is referenced by:  ax12  2292  axc11n  2295  axc11nOLD  2296  axc11nOLDOLD  2297  axc11nALT  2298  axc11  2302  hbae  2303  dral1  2313  dral1ALT  2314  axpowndlem3  9300  axc11n11r  31860  bj-ax12v3ALT  31863  bj-axc11v  31935  bj-dral1v  31936  bj-hbaeb2  31993
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