Theorem bj-ax89 34015

Description: A theorem which could be used as sole axiom for the non-logical predicate instead of ax-8 2115 and ax-9 2123. Indeed, it is implied over propositional calculus by the conjunction of ax-8 2115 and ax-9 2123, as proved here. In the other direction, one can prove ax-8 2115 (respectively ax-9 2123) from bj-ax89 34015 by using mpan2 689 (respectively mpan 688) and equid 2018. TODO: move to main part. (Contributed by BJ, 3-Oct-2019.)

Assertion

Ref	Expression
bj-ax89	⊢ ((𝑥 = 𝑦 ∧ 𝑧 = 𝑡) → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑡))

Proof of Theorem bj-ax89

Step	Hyp	Ref	Expression
1		ax8 2119	. 2 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧))
2		ax9 2127	. 2 ⊢ (𝑧 = 𝑡 → (𝑦 ∈ 𝑧 → 𝑦 ∈ 𝑡))
3	1, 2	sylan9 510	1 ⊢ ((𝑥 = 𝑦 ∧ 𝑧 = 𝑡) → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑡))

Syntax hints:   → wi 4   ∧ wa 398

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1780

This theorem is referenced by: (None)