Theorem ax9 2113

Description: Proof of ax-9 2109 from ax9v1 2111 and ax9v2 2112, proving sufficiency of the conjunction of the latter two weakened versions of ax9v 2110, which is itself a weakened version of ax-9 2109. (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 2025	. 2 ⊢ (𝑥 = 𝑦 ↔ ∃𝑡(𝑡 = 𝑥 ∧ 𝑡 = 𝑦))
2		ax9v2 2112	. . . . 5 ⊢ (𝑥 = 𝑡 → (𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑡))
3	2	equcoms 2016	. . . 4 ⊢ (𝑡 = 𝑥 → (𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑡))
4		ax9v1 2111	. . . 4 ⊢ (𝑡 = 𝑦 → (𝑧 ∈ 𝑡 → 𝑧 ∈ 𝑦))
5	3, 4	sylan9 506	. . 3 ⊢ ((𝑡 = 𝑥 ∧ 𝑡 = 𝑦) → (𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦))
6	5	exlimiv 1926	. 2 ⊢ (∃𝑡(𝑡 = 𝑥 ∧ 𝑡 = 𝑦) → (𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦))
7	1, 6	sylbi 216	1 ⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦))

Syntax hints:   → wi 4   ∧ wa 394  ∃wex 1774

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-9 2109

This theorem depends on definitions:  df-bi 206  df-an 395  df-ex 1775

This theorem is referenced by:  elequ2  2114  elALT2  5373  fv3  6919  elirrv  9639  bj-ax89  36382  axc11next  44080