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Theorem axc11r 2185
Description: Same as axc11 2312 but with reversed antecedent. Note the use of ax-12 2045 (and not merely ax12v 2046). (Contributed by NM, 25-Jul-2015.)
Assertion
Ref Expression
axc11r (∀𝑦 𝑦 = 𝑥 → (∀𝑥𝜑 → ∀𝑦𝜑))

Proof of Theorem axc11r
StepHypRef Expression
1 ax-12 2045 . . 3 (𝑦 = 𝑥 → (∀𝑥𝜑 → ∀𝑦(𝑦 = 𝑥𝜑)))
21sps 2053 . 2 (∀𝑦 𝑦 = 𝑥 → (∀𝑥𝜑 → ∀𝑦(𝑦 = 𝑥𝜑)))
3 pm2.27 42 . . 3 (𝑦 = 𝑥 → ((𝑦 = 𝑥𝜑) → 𝜑))
43al2imi 1741 . 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 1479
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-12 2045
This theorem depends on definitions:  df-bi 197  df-ex 1703
This theorem is referenced by:  ax12  2302  axc11n  2305  axc11nOLD  2306  axc11nOLDOLD  2307  axc11nALT  2308  axc11  2312  hbae  2313  dral1  2323  dral1ALT  2324  axpowndlem3  9406  axc11n11r  32648  bj-ax12v3ALT  32651  bj-axc11v  32722  bj-dral1v  32723  bj-hbaeb2  32780
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