Theorem bj-hblem 32043

Description: Remove dependency on ax-ext 2590 (and df-cleq 2603) from hblem 2718. (Contributed by BJ, 24-Jun-2019.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
bj-hblem.1	⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴)

Assertion

Ref	Expression
bj-hblem	⊢ (𝑧 ∈ 𝐴 → ∀𝑥 𝑧 ∈ 𝐴)

Distinct variable groups:   𝑦,𝐴   𝑥,𝑧

Allowed substitution hints:   𝐴(𝑥,𝑧)

Proof of Theorem bj-hblem

Step	Hyp	Ref	Expression
1		bj-hblem.1	. . 3 ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴)
2	1	hbsb 2429	. 2 ⊢ ([𝑧 / 𝑦]𝑦 ∈ 𝐴 → ∀𝑥[𝑧 / 𝑦]𝑦 ∈ 𝐴)
3		bj-clelsb3 32042	. 2 ⊢ ([𝑧 / 𝑦]𝑦 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)
4	3	albii 1737	. 2 ⊢ (∀𝑥[𝑧 / 𝑦]𝑦 ∈ 𝐴 ↔ ∀𝑥 𝑧 ∈ 𝐴)
5	2, 3, 4	3imtr3i 279	1 ⊢ (𝑧 ∈ 𝐴 → ∀𝑥 𝑧 ∈ 𝐴)

Syntax hints:   → wi 4  ∀wal 1473  [wsb 1867   ∈ wcel 1977

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  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clel 2606

This theorem is referenced by:  bj-nfcrii  32045