Theorem ax9ALT 2817

Description: Proof of ax-9 2124 from Tarski's FOL and dfcleq 2815. For a version not using ax-8 2116 either, see bj-ax9 34219. This shows that dfcleq 2815 is too powerful to be used as a definition instead of df-cleq 2814. Note that ax-ext 2793 is also a direct consequence of dfcleq 2815 (as an instance of its forward implication). (Contributed by BJ, 24-Jun-2019.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
ax9ALT	⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦))

Proof of Theorem ax9ALT

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfcleq 2815	. . . 4 ⊢ (𝑥 = 𝑦 ↔ ∀𝑡(𝑡 ∈ 𝑥 ↔ 𝑡 ∈ 𝑦))
2	1	biimpi 218	. . 3 ⊢ (𝑥 = 𝑦 → ∀𝑡(𝑡 ∈ 𝑥 ↔ 𝑡 ∈ 𝑦))
3		biimp 217	. . 3 ⊢ ((𝑡 ∈ 𝑥 ↔ 𝑡 ∈ 𝑦) → (𝑡 ∈ 𝑥 → 𝑡 ∈ 𝑦))
4	2, 3	sylg 1823	. 2 ⊢ (𝑥 = 𝑦 → ∀𝑡(𝑡 ∈ 𝑥 → 𝑡 ∈ 𝑦))
5		ax8 2120	. . . . 5 ⊢ (𝑧 = 𝑡 → (𝑧 ∈ 𝑥 → 𝑡 ∈ 𝑥))
6	5	equcoms 2027	. . . 4 ⊢ (𝑡 = 𝑧 → (𝑧 ∈ 𝑥 → 𝑡 ∈ 𝑥))
7		ax8 2120	. . . 4 ⊢ (𝑡 = 𝑧 → (𝑡 ∈ 𝑦 → 𝑧 ∈ 𝑦))
8	6, 7	imim12d 81	. . 3 ⊢ (𝑡 = 𝑧 → ((𝑡 ∈ 𝑥 → 𝑡 ∈ 𝑦) → (𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦)))
9	8	spimvw 2002	. 2 ⊢ (∀𝑡(𝑡 ∈ 𝑥 → 𝑡 ∈ 𝑦) → (𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦))
10	4, 9	syl 17	1 ⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦))

Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781  df-cleq 2814

This theorem is referenced by: (None)