Theorem wl-axc11rc11 37955

Description: Proving axc11r 2376 from axc11 2438. The hypotheses are two instances of axc11 2438 used in the proof here. Some systems introduce axc11 2438 as an axiom, see for example System S2 in https://us.metamath.org/downloads/finiteaxiom.pdf 2438.

By contrast, this database sees the variant axc11r 2376, directly derived from ax-12 2189, as foundational. Later axc11 2438 is proven somewhat trickily, requiring ax-10 2152 and ax-13 2380, 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 2024	. . 3 ⊢ (𝑦 = 𝑥 → 𝑥 = 𝑦)
4	3	alimi 1818	. 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 1545

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015

This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1787

This theorem is referenced by: (None)