Theorem bj-ax89 32667

Description: A theorem which could be used as sole axiom for the non-logical predicate instead of ax-8 1992 and ax-9 1999. Indeed, it is implied over propositional calculus by the conjunction of ax-8 1992 and ax-9 1999, as proved here. In the other direction, one can prove ax-8 1992 (respectively ax-9 1999) from bj-ax89 32667 by using mpan2 707 ( respectively mpan 706) and equid 1939. (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 1996	. 2 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧))
2		ax9 2003	. 2 ⊢ (𝑧 = 𝑡 → (𝑦 ∈ 𝑧 → 𝑦 ∈ 𝑡))
3	1, 2	sylan9 689	1 ⊢ ((𝑥 = 𝑦 ∧ 𝑧 = 𝑡) → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑡))

Syntax hints:   → wi 4   ∧ wa 384

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705

This theorem is referenced by: (None)