Theorem axc11r 2366

Description: Same as axc11 2430 but with reversed antecedent. Note the use of ax-12 2171 (and not merely ax12v 2172 as in axc11rv 2257).

This theorem is mostly used to eliminate conditions requiring set variables be distinct (cf. cbvaev 2056 and aecom 2427, 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 2372 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 2171	. . 3 ⊢ (𝑦 = 𝑥 → (∀𝑥𝜑 → ∀𝑦(𝑦 = 𝑥 → 𝜑)))
2	1	sps 2178	. 2 ⊢ (∀𝑦 𝑦 = 𝑥 → (∀𝑥𝜑 → ∀𝑦(𝑦 = 𝑥 → 𝜑)))
3		pm2.27 42	. . 3 ⊢ (𝑦 = 𝑥 → ((𝑦 = 𝑥 → 𝜑) → 𝜑))
4	3	al2imi 1818	. 2 ⊢ (∀𝑦 𝑦 = 𝑥 → (∀𝑦(𝑦 = 𝑥 → 𝜑) → ∀𝑦𝜑))
5	2, 4	syld 47	1 ⊢ (∀𝑦 𝑦 = 𝑥 → (∀𝑥𝜑 → ∀𝑦𝜑))

Syntax hints:   → wi 4  ∀wal 1537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-12 2171

This theorem depends on definitions:  df-bi 206  df-ex 1783

This theorem is referenced by:  ax12  2423  axc11n  2426  axc11  2430  hbae  2431  dral1  2439  dral1ALT  2440  sb4a  2484  axpowndlem3  10355  axc11n11r  34865  bj-ax12v3ALT  34868  bj-hbaeb2  35001