Theorem bj-ax89 33173

Description: A theorem which could be used as sole axiom for the non-logical predicate instead of ax-8 2159 and ax-9 2166. Indeed, it is implied over propositional calculus by the conjunction of ax-8 2159 and ax-9 2166, as proved here. In the other direction, one can prove ax-8 2159 (respectively ax-9 2166) from bj-ax89 33173 by using mpan2 683 ( respectively mpan 682) and equid 2111. (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 2163	. 2 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧))
2		ax9 2170	. 2 ⊢ (𝑧 = 𝑡 → (𝑦 ∈ 𝑧 → 𝑦 ∈ 𝑡))
3	1, 2	sylan9 504	1 ⊢ ((𝑥 = 𝑦 ∧ 𝑧 = 𝑡) → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑡))

Syntax hints:   → wi 4   ∧ wa 385

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166

This theorem depends on definitions:  df-bi 199  df-an 386  df-ex 1876

This theorem is referenced by: (None)