Theorem bj-ax89 34124

Description: A theorem which could be used as sole axiom for the non-logical predicate instead of ax-8 2113 and ax-9 2121. Indeed, it is implied over propositional calculus by the conjunction of ax-8 2113 and ax-9 2121, as proved here. In the other direction, one can prove ax-8 2113 (respectively ax-9 2121) from bj-ax89 34124 by using mpan2 690 (respectively mpan 689) and equid 2019. 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 2117	. 2 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧))
2		ax9 2125	. 2 ⊢ (𝑧 = 𝑡 → (𝑦 ∈ 𝑧 → 𝑦 ∈ 𝑡))
3	1, 2	sylan9 511	1 ⊢ ((𝑥 = 𝑦 ∧ 𝑧 = 𝑡) → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑡))

Syntax hints:   → wi 4   ∧ wa 399

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782

This theorem is referenced by: (None)