Theorem bj-ax89 34786

Description: A theorem which could be used as sole axiom for the non-logical predicate instead of ax-8 2110 and ax-9 2118. Indeed, it is implied over propositional calculus by the conjunction of ax-8 2110 and ax-9 2118, as proved here. In the other direction, one can prove ax-8 2110 (respectively ax-9 2118) from bj-ax89 34786 by using mpan2 687 (respectively mpan 686) and equid 2016. 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 2114	. 2 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧))
2		ax9 2122	. 2 ⊢ (𝑧 = 𝑡 → (𝑦 ∈ 𝑧 → 𝑦 ∈ 𝑡))
3	1, 2	sylan9 507	1 ⊢ ((𝑥 = 𝑦 ∧ 𝑧 = 𝑡) → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑡))

Syntax hints:   → wi 4   ∧ wa 395

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1784

This theorem is referenced by: (None)