Theorem axc11r 2400

Description: Same as axc11 2462 but with reversed antecedent. Note the use of ax-12 2213 (and not merely ax12v 2214 as in axc11rv 2301).

This theorem is mostly used to eliminate conditions requiring set variables be distinct (cf. cbvaev 2085 and aecom 2459, 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 2404 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 2213	. . 3 ⊢ (𝑦 = 𝑥 → (∀𝑥𝜑 → ∀𝑦(𝑦 = 𝑥 → 𝜑)))
2	1	sps 2221	. 2 ⊢ (∀𝑦 𝑦 = 𝑥 → (∀𝑥𝜑 → ∀𝑦(𝑦 = 𝑥 → 𝜑)))
3		pm2.27 43	. . 3 ⊢ (𝑦 = 𝑥 → ((𝑦 = 𝑥 → 𝜑) → 𝜑))
4	3	al2imi 1845	. 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 1568

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-12 2213

This theorem depends on definitions:  df-bi 210  df-ex 1810

This theorem is referenced by:  ax12  2455  axc11n  2458  axc11  2462  hbae  2463  dral1  2471  dral1ALT  2472  sb4a  2512  axpowndlem3  10585  axpowg2  35538  axpowg3  35539  axc11n11r  37286  bj-ax12v3ALT  37289  bj-hbaeb2  37431