Theorem bj-ax8 33458

Description: Proof of ax-8 2108 from df-clel 2773 (and FOL). This shows that df-clel 2773 is "too powerful". A possible definition is given by bj-df-clel 33459. (Contributed by BJ, 27-Jun-2019.) Also a direct consequence of eleq1w 2841, 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 2072	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑢 = 𝑥 ↔ 𝑢 = 𝑦))
2	1	anbi1d 623	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝑢 = 𝑥 ∧ 𝑢 ∈ 𝑧) ↔ (𝑢 = 𝑦 ∧ 𝑢 ∈ 𝑧)))
3	2	exbidv 1964	. . 3 ⊢ (𝑥 = 𝑦 → (∃𝑢(𝑢 = 𝑥 ∧ 𝑢 ∈ 𝑧) ↔ ∃𝑢(𝑢 = 𝑦 ∧ 𝑢 ∈ 𝑧)))
4		df-clel 2773	. . 3 ⊢ (𝑥 ∈ 𝑧 ↔ ∃𝑢(𝑢 = 𝑥 ∧ 𝑢 ∈ 𝑧))
5		df-clel 2773	. . 3 ⊢ (𝑦 ∈ 𝑧 ↔ ∃𝑢(𝑢 = 𝑦 ∧ 𝑢 ∈ 𝑧))
6	3, 4, 5	3bitr4g 306	. 2 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑧 ↔ 𝑦 ∈ 𝑧))
7	6	biimpd 221	1 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧))

Syntax hints:   → wi 4   ∧ wa 386  ∃wex 1823

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054

This theorem depends on definitions:  df-bi 199  df-an 387  df-ex 1824  df-clel 2773

This theorem is referenced by: (None)