Theorem ax9 2128

Description: Proof of ax-9 2124 from ax9v1 2126 and ax9v2 2127, proving sufficiency of the conjunction of the latter two weakened versions of ax9v 2125, which is itself a weakened version of ax-9 2124. (Contributed by BJ, 7-Dec-2020.) (Proof shortened by Wolf Lammen, 11-Apr-2021.)

Assertion

Ref	Expression
ax9	⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦))

Proof of Theorem ax9

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		equvinv 2031	. 2 ⊢ (𝑥 = 𝑦 ↔ ∃𝑡(𝑡 = 𝑥 ∧ 𝑡 = 𝑦))
2		ax9v2 2127	. . . . 5 ⊢ (𝑥 = 𝑡 → (𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑡))
3	2	equcoms 2022	. . . 4 ⊢ (𝑡 = 𝑥 → (𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑡))
4		ax9v1 2126	. . . 4 ⊢ (𝑡 = 𝑦 → (𝑧 ∈ 𝑡 → 𝑧 ∈ 𝑦))
5	3, 4	sylan9 507	. . 3 ⊢ ((𝑡 = 𝑥 ∧ 𝑡 = 𝑦) → (𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦))
6	5	exlimiv 1932	. 2 ⊢ (∃𝑡(𝑡 = 𝑥 ∧ 𝑡 = 𝑦) → (𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦))
7	1, 6	sylbi 217	1 ⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦))

Syntax hints:   → wi 4   ∧ wa 395  ∃wex 1781

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-9 2124

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782

This theorem is referenced by:  elequ2  2129  elALT2  5316  fv3  6860  elirrvOLD  9515  in-ax8  36437  ss-ax8  36438  bj-ax89  36917  axc11next  44756