Theorem bj-ralvw 34991

Description: A weak version of ralv 3446 not using ax-ext 2709 (nor df-cleq 2730, df-clel 2817, df-v 3424), and only core FOL axioms. See also bj-rexvw 34992. The analogues for reuv 3448 and rmov 3449 are not proved. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
bj-ralvw.1	⊢ 𝜓

Assertion

Ref	Expression
bj-ralvw	⊢ (∀𝑥 ∈ {𝑦 ∣ 𝜓}𝜑 ↔ ∀𝑥𝜑)

Proof of Theorem bj-ralvw

Step	Hyp	Ref	Expression
1		df-ral 3068	. 2 ⊢ (∀𝑥 ∈ {𝑦 ∣ 𝜓}𝜑 ↔ ∀𝑥(𝑥 ∈ {𝑦 ∣ 𝜓} → 𝜑))
2		bj-ralvw.1	. . . . 5 ⊢ 𝜓
3	2	vexw 2721	. . . 4 ⊢ 𝑥 ∈ {𝑦 ∣ 𝜓}
4	3	a1bi 362	. . 3 ⊢ (𝜑 ↔ (𝑥 ∈ {𝑦 ∣ 𝜓} → 𝜑))
5	4	albii 1823	. 2 ⊢ (∀𝑥𝜑 ↔ ∀𝑥(𝑥 ∈ {𝑦 ∣ 𝜓} → 𝜑))
6	1, 5	bitr4i 277	1 ⊢ (∀𝑥 ∈ {𝑦 ∣ 𝜓}𝜑 ↔ ∀𝑥𝜑)

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1537   ∈ wcel 2108  {cab 2715  ∀wral 3063

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813

This theorem depends on definitions:  df-bi 206  df-sb 2069  df-clab 2716  df-ral 3068

This theorem is referenced by: (None)