Theorem bj-ax9 34211

Description: Proof of ax-9 2120 from Tarski's FOL=, sp 2178, dfcleq 2815 and ax-ext 2793 (with two extra disjoint variable conditions on 𝑥, 𝑧 and 𝑦, 𝑧). See ax9ALT 2817 for a more general version, proved using also ax-8 2112. (Contributed by BJ, 24-Jun-2019.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
bj-ax9	⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦))

Distinct variable groups:   𝑥,𝑧   𝑦,𝑧

Proof of Theorem bj-ax9

Step	Hyp	Ref	Expression
1		dfcleq 2815	. 2 ⊢ (𝑥 = 𝑦 ↔ ∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦))
2		biimp 217	. . 3 ⊢ ((𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → (𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦))
3	2	sps 2180	. 2 ⊢ (∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → (𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦))
4	1, 3	sylbi 219	1 ⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦))

Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1531

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-9 2120  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1777  df-cleq 2814

This theorem is referenced by: (None)