Theorem axc11r 2398

Description: Same as axc11 2460 but with reversed antecedent. Note the use of ax-12 2211 (and not merely ax12v 2212 as in axc11rv 2299).

This theorem is mostly used to eliminate conditions requiring set variables be distinct (cf. cbvaev 2074 and aecom 2457, 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 2402 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 2211	. . 3 ⊢ (𝑦 = 𝑥 → (∀𝑥𝜑 → ∀𝑦(𝑦 = 𝑥 → 𝜑)))
2	1	sps 2219	. 2 ⊢ (∀𝑦 𝑦 = 𝑥 → (∀𝑥𝜑 → ∀𝑦(𝑦 = 𝑥 → 𝜑)))
3		pm2.27 42	. . 3 ⊢ (𝑦 = 𝑥 → ((𝑦 = 𝑥 → 𝜑) → 𝜑))
4	3	al2imi 1834	. 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 1557

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-12 2211

This theorem depends on definitions:  df-bi 209  df-ex 1799

This theorem is referenced by:  ax12  2453  axc11n  2456  axc11  2460  hbae  2461  dral1  2469  dral1ALT  2470  sb4a  2510  axpowndlem3  10554  axpowg2  35407  axpowg3  35408  axc11n11r  37122  bj-ax12v3ALT  37125  bj-hbaeb2  37267