Theorem bj-ralvw 32059

Description: A weak version of ralv 3192 not using ax-ext 2590 (nor df-cleq 2603, df-clel 2606, df-v 3175), but using ax-13 2234. For the sake of illustration, the next theorem bj-rexvwv 32060, a weak version of rexv 3193, has a dv condition and avoids dependency on ax-13 2234, while the analogues for reuv 3194 and rmov 3195 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 2901	. 2 ⊢ (∀𝑥 ∈ {𝑦 ∣ 𝜓}𝜑 ↔ ∀𝑥(𝑥 ∈ {𝑦 ∣ 𝜓} → 𝜑))
2		bj-ralvw.1	. . . . 5 ⊢ 𝜓
3	2	bj-vexw 32049	. . . 4 ⊢ 𝑥 ∈ {𝑦 ∣ 𝜓}
4	3	a1bi 351	. . 3 ⊢ (𝜑 ↔ (𝑥 ∈ {𝑦 ∣ 𝜓} → 𝜑))
5	4	albii 1737	. 2 ⊢ (∀𝑥𝜑 ↔ ∀𝑥(𝑥 ∈ {𝑦 ∣ 𝜓} → 𝜑))
6	1, 5	bitr4i 266	1 ⊢ (∀𝑥 ∈ {𝑦 ∣ 𝜓}𝜑 ↔ ∀𝑥𝜑)

Syntax hints:   → wi 4   ↔ wb 195  ∀wal 1473   ∈ wcel 1977  {cab 2596  ∀wral 2896

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-12 2034  ax-13 2234

This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-sb 1868  df-clab 2597  df-ral 2901

This theorem is referenced by: (None)