Theorem bj-ax9 33015

Description: Proof of ax-9 2039 from Tarski's FOL=, sp 2091, df-cleq 2644 and ax-ext 2631 (with two extra dv conditions on 𝑥, 𝑧 and 𝑦, 𝑧). For a version without these dv conditions, see bj-ax9-2 33016. This shows that df-cleq 2644 is "too powerful". A possible definition is given by bj-df-cleq 33018. (Contributed by BJ, 24-Jun-2019.) (Proof modification 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

Dummy variables 𝑤 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-ext 2631	. . 3 ⊢ (∀𝑧(𝑧 ∈ 𝑢 ↔ 𝑧 ∈ 𝑤) → 𝑢 = 𝑤)
2	1	df-cleq 2644	. 2 ⊢ (𝑥 = 𝑦 ↔ ∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦))
3		biimp 205	. . 3 ⊢ ((𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → (𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦))
4	3	sps 2093	. 2 ⊢ (∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → (𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦))
5	2, 4	sylbi 207	1 ⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦))

Syntax hints:   → wi 4   ↔ wb 196  ∀wal 1521

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-12 2087  ax-ext 2631

This theorem depends on definitions:  df-bi 197  df-ex 1745  df-cleq 2644

This theorem is referenced by: (None)