Theorem bj-ax89 31854

Description: A theorem which could be used as sole axiom for the non-logical predicate instead of ax-8 1979 and ax-9 1986. Indeed, it is implied over propositional calculus by the conjunction of ax-8 1979 and ax-9 1986, as proved here. In the other direction, one can prove ax-8 1979 (respectively ax-9 1986) from bj-ax89 31854 by using mpan2 703 ( respectively mpan 702) and equid 1926. (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 1983	. 2 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧))
2		ax9 1990	. 2 ⊢ (𝑧 = 𝑡 → (𝑦 ∈ 𝑧 → 𝑦 ∈ 𝑡))
3	1, 2	sylan9 687	1 ⊢ ((𝑥 = 𝑦 ∧ 𝑧 = 𝑡) → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑡))

Syntax hints:   → wi 4   ∧ wa 383

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986

This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696

This theorem is referenced by: (None)