Theorem wl-axc11rc11 37585

Description: Proving axc11r 2370 from axc11 2434. The hypotheses are two instances of axc11 2434 used in the proof here. Some systems introduce axc11 2434 as an axiom, see for example System S2 in https://us.metamath.org/downloads/finiteaxiom.pdf 2434.

By contrast, this database sees the variant axc11r 2370, directly derived from ax-12 2176, as foundational. Later axc11 2434 is proven somewhat trickily, requiring ax-10 2140 and ax-13 2376, see its proof. (Contributed by Wolf Lammen, 18-Jul-2023.)

Hypotheses

Ref	Expression
wl-axc11rc11.1	⊢ (∀𝑦 𝑦 = 𝑥 → (∀𝑦 𝑦 = 𝑥 → ∀𝑥 𝑦 = 𝑥))
wl-axc11rc11.2	⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑))

Assertion

Ref	Expression
wl-axc11rc11	⊢ (∀𝑦 𝑦 = 𝑥 → (∀𝑥𝜑 → ∀𝑦𝜑))

Proof of Theorem wl-axc11rc11

Step	Hyp	Ref	Expression
1		wl-axc11rc11.1	. . 3 ⊢ (∀𝑦 𝑦 = 𝑥 → (∀𝑦 𝑦 = 𝑥 → ∀𝑥 𝑦 = 𝑥))
2	1	pm2.43i 52	. 2 ⊢ (∀𝑦 𝑦 = 𝑥 → ∀𝑥 𝑦 = 𝑥)
3		equcomi 2015	. . 3 ⊢ (𝑦 = 𝑥 → 𝑥 = 𝑦)
4	3	alimi 1810	. 2 ⊢ (∀𝑥 𝑦 = 𝑥 → ∀𝑥 𝑥 = 𝑦)
5		wl-axc11rc11.2	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑))
6	2, 4, 5	3syl 18	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 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1779

This theorem is referenced by: (None)