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Theorem axc11r 2364
Description: Same as axc11 2428 but with reversed antecedent. Note the use of ax-12 2170 (and not merely ax12v 2171 as in axc11rv 2255).

This theorem is mostly used to eliminate conditions requiring set variables be distinct (cf. cbvaev 2055 and aecom 2425, for example) in proofs. In practice, theorems beyond elementary set theory do not really benefit from such eliminations. As of 2024, it is used in conjunction with ax-13 2370 only, and like that, it should be applied only in niches where indispensable. (Contributed by NM, 25-Jul-2015.)

Assertion
Ref Expression
axc11r (∀𝑦 𝑦 = 𝑥 → (∀𝑥𝜑 → ∀𝑦𝜑))

Proof of Theorem axc11r
StepHypRef Expression
1 ax-12 2170 . . 3 (𝑦 = 𝑥 → (∀𝑥𝜑 → ∀𝑦(𝑦 = 𝑥𝜑)))
21sps 2177 . 2 (∀𝑦 𝑦 = 𝑥 → (∀𝑥𝜑 → ∀𝑦(𝑦 = 𝑥𝜑)))
3 pm2.27 42 . . 3 (𝑦 = 𝑥 → ((𝑦 = 𝑥𝜑) → 𝜑))
43al2imi 1816 . 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 1538
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-12 2170
This theorem depends on definitions:  df-bi 206  df-ex 1781
This theorem is referenced by:  ax12  2421  axc11n  2424  axc11  2428  hbae  2429  dral1  2437  dral1ALT  2438  sb4a  2478  axpowndlem3  10600  axc11n11r  36025  bj-ax12v3ALT  36028  bj-hbaeb2  36160
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