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Theorem axc11r 2332
Description: Same as axc11 2456 but with reversed antecedent. Note the use of ax-12 2196 (and not merely ax12v 2197). (Contributed by NM, 25-Jul-2015.)
Assertion
Ref Expression
axc11r (∀𝑦 𝑦 = 𝑥 → (∀𝑥𝜑 → ∀𝑦𝜑))

Proof of Theorem axc11r
StepHypRef Expression
1 ax-12 2196 . . 3 (𝑦 = 𝑥 → (∀𝑥𝜑 → ∀𝑦(𝑦 = 𝑥𝜑)))
21sps 2202 . 2 (∀𝑦 𝑦 = 𝑥 → (∀𝑥𝜑 → ∀𝑦(𝑦 = 𝑥𝜑)))
3 pm2.27 42 . . 3 (𝑦 = 𝑥 → ((𝑦 = 𝑥𝜑) → 𝜑))
43al2imi 1892 . 2 (∀𝑦 𝑦 = 𝑥 → (∀𝑦(𝑦 = 𝑥𝜑) → ∀𝑦𝜑))
52, 4syld 47 1 (∀𝑦 𝑦 = 𝑥 → (∀𝑥𝜑 → ∀𝑦𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1630
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-12 2196
This theorem depends on definitions:  df-bi 197  df-ex 1854
This theorem is referenced by:  ax12  2449  axc11n  2451  axc11nOLD  2452  axc11  2456  hbae  2457  dral1  2465  dral1ALT  2466  axpowndlem3  9613  axc11n11r  32979  bj-ax12v3ALT  32982  bj-axc11v  33053  bj-dral1v  33054  bj-hbaeb2  33111
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