Theorem axc11r 2400

Description: Same as axc11 2462 but with reversed antecedent. Note the use of ax-12 2216 (and not merely ax12v 2217 as in axc11rv 2303).

This theorem is mostly used to eliminate conditions requiring set variables be distinct (cf. cbvaev 2090 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 2216	. . 3 ⊢ (𝑦 = 𝑥 → (∀𝑥𝜑 → ∀𝑦(𝑦 = 𝑥 → 𝜑)))
2	1	sps 2225	. 2 ⊢ (∀𝑦 𝑦 = 𝑥 → (∀𝑥𝜑 → ∀𝑦(𝑦 = 𝑥 → 𝜑)))
3		pm2.27 43	. . 3 ⊢ (𝑦 = 𝑥 → ((𝑦 = 𝑥 → 𝜑) → 𝜑))
4	3	al2imi 1848	. 2 ⊢ (∀𝑦 𝑦 = 𝑥 → (∀𝑦(𝑦 = 𝑥 → 𝜑) → ∀𝑦𝜑))
5	2, 4	syld 48	1 ⊢ (∀𝑦 𝑦 = 𝑥 → (∀𝑥𝜑 → ∀𝑦𝜑))

This proof depends on syntax axioms:   → wi 4  ∀wal 1568

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-12 2216

This proof depends on definitions:  df-bi 210  df-ex 1813

This theorem is used by:  ax12  2455  axc11n  2458  axc11  2462  hbae  2463  dral1  2471  dral1ALT  2472  sb4a  2512  axpowndlem3  10711  axpowg2  35915  axpowg3  35916  axc11n11r  37763  bj-ax12v3ALT  37766  bj-hbaeb2  37908