Theorem axc11r 2402

Description: Same as axc11 2464 but with reversed antecedent. Note the use of ax-12 2215 (and not merely ax12v 2216 as in axc11rv 2303).

This theorem is mostly used to eliminate conditions requiring set variables be distinct (cf. cbvaev 2078 and aecom 2461, 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 2406 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

Step	Hyp	Ref	Expression
1		ax-12 2215	. . 3 ⊢ (𝑦 = 𝑥 → (∀𝑥𝜑 → ∀𝑦(𝑦 = 𝑥 → 𝜑)))
2	1	sps 2223	. 2 ⊢ (∀𝑦 𝑦 = 𝑥 → (∀𝑥𝜑 → ∀𝑦(𝑦 = 𝑥 → 𝜑)))
3		pm2.27 43	. . 3 ⊢ (𝑦 = 𝑥 → ((𝑦 = 𝑥 → 𝜑) → 𝜑))
4	3	al2imi 1838	. 2 ⊢ (∀𝑦 𝑦 = 𝑥 → (∀𝑦(𝑦 = 𝑥 → 𝜑) → ∀𝑦𝜑))
5	2, 4	syld 48	1 ⊢ (∀𝑦 𝑦 = 𝑥 → (∀𝑥𝜑 → ∀𝑦𝜑))

Syntax hints:   → wi 4  ∀wal 1561

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-12 2215

This theorem depends on definitions:  df-bi 210  df-ex 1803

This theorem is referenced by:  ax12  2457  axc11n  2460  axc11  2464  hbae  2465  dral1  2473  dral1ALT  2474  sb4a  2514  axpowndlem3  10572  axpowg2  35455  axpowg3  35456  axc11n11r  37170  bj-ax12v3ALT  37173  bj-hbaeb2  37315