Theorem bj-ax8 32080

Description: Proof of ax-8 1979 from df-clel 2606 (and FOL). This shows that df-clel 2606 is "too powerful". A possible definition is given by bj-df-clel 32081. (Contributed by BJ, 27-Jun-2019.) Also a direct consequence of bj-eleq1w 32040, which has essentially the same proof. (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-ax8	⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧))

Proof of Theorem bj-ax8

Dummy variable 𝑢 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		equequ2 1940	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑢 = 𝑥 ↔ 𝑢 = 𝑦))
2	1	anbi1d 737	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝑢 = 𝑥 ∧ 𝑢 ∈ 𝑧) ↔ (𝑢 = 𝑦 ∧ 𝑢 ∈ 𝑧)))
3	2	exbidv 1837	. . 3 ⊢ (𝑥 = 𝑦 → (∃𝑢(𝑢 = 𝑥 ∧ 𝑢 ∈ 𝑧) ↔ ∃𝑢(𝑢 = 𝑦 ∧ 𝑢 ∈ 𝑧)))
4		df-clel 2606	. . 3 ⊢ (𝑥 ∈ 𝑧 ↔ ∃𝑢(𝑢 = 𝑥 ∧ 𝑢 ∈ 𝑧))
5		df-clel 2606	. . 3 ⊢ (𝑦 ∈ 𝑧 ↔ ∃𝑢(𝑢 = 𝑦 ∧ 𝑢 ∈ 𝑧))
6	3, 4, 5	3bitr4g 302	. 2 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑧 ↔ 𝑦 ∈ 𝑧))
7	6	biimpd 218	1 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧))

Syntax hints:   → wi 4   ∧ wa 383  ∃wex 1695

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

This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-clel 2606

This theorem is referenced by: (None)