Theorem ax8 2112

Description: Proof of ax-8 2108 from ax8v1 2110 and ax8v2 2111, proving sufficiency of the conjunction of the latter two weakened versions of ax8v 2109, which is itself a weakened version of ax-8 2108. (Contributed by BJ, 7-Dec-2020.) (Proof shortened by Wolf Lammen, 11-Apr-2021.)

Assertion

Ref	Expression
ax8	⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧))

Proof of Theorem ax8

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		equvinv 2032	. 2 ⊢ (𝑥 = 𝑦 ↔ ∃𝑡(𝑡 = 𝑥 ∧ 𝑡 = 𝑦))
2		ax8v2 2111	. . . . 5 ⊢ (𝑥 = 𝑡 → (𝑥 ∈ 𝑧 → 𝑡 ∈ 𝑧))
3	2	equcoms 2023	. . . 4 ⊢ (𝑡 = 𝑥 → (𝑥 ∈ 𝑧 → 𝑡 ∈ 𝑧))
4		ax8v1 2110	. . . 4 ⊢ (𝑡 = 𝑦 → (𝑡 ∈ 𝑧 → 𝑦 ∈ 𝑧))
5	3, 4	sylan9 508	. . 3 ⊢ ((𝑡 = 𝑥 ∧ 𝑡 = 𝑦) → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧))
6	5	exlimiv 1933	. 2 ⊢ (∃𝑡(𝑡 = 𝑥 ∧ 𝑡 = 𝑦) → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧))
7	1, 6	sylbi 216	1 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧))

Syntax hints:   → wi 4   ∧ wa 396  ∃wex 1782

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1783

This theorem is referenced by:  elequ1  2113  ax9ALT  2733  elALT2  5292  axextdfeq  33773  ax8dfeq  33774  exnel  33778  bj-ax89  34859