Theorem bj-rexcom4b 32511

Description: Remove from rexcom4b 3218 dependency on ax-ext 2606 and ax-13 2250 (and on df-or 385, df-cleq 2619, df-nfc 2756, df-v 3193). The hypothesis uses 𝑉 instead of V (see bj-isseti 32503 for the motivation). Use bj-rexcom4bv 32510 instead when sufficient (in particular when 𝑉 is substituted for V). (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
bj-rexcom4b.1	⊢ 𝐵 ∈ 𝑉

Assertion

Ref	Expression
bj-rexcom4b	⊢ (∃𝑥∃𝑦 ∈ 𝐴 (𝜑 ∧ 𝑥 = 𝐵) ↔ ∃𝑦 ∈ 𝐴 𝜑)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑦   𝜑,𝑥

Allowed substitution hints:   𝜑(𝑦)   𝐴(𝑦)   𝐵(𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem bj-rexcom4b

Step	Hyp	Ref	Expression
1		bj-rexcom4a 32509	. 2 ⊢ (∃𝑥∃𝑦 ∈ 𝐴 (𝜑 ∧ 𝑥 = 𝐵) ↔ ∃𝑦 ∈ 𝐴 (𝜑 ∧ ∃𝑥 𝑥 = 𝐵))
2		bj-rexcom4b.1	. . . . 5 ⊢ 𝐵 ∈ 𝑉
3	2	bj-isseti 32503	. . . 4 ⊢ ∃𝑥 𝑥 = 𝐵
4	3	biantru 526	. . 3 ⊢ (𝜑 ↔ (𝜑 ∧ ∃𝑥 𝑥 = 𝐵))
5	4	rexbii 3039	. 2 ⊢ (∃𝑦 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 (𝜑 ∧ ∃𝑥 𝑥 = 𝐵))
6	1, 5	bitr4i 267	1 ⊢ (∃𝑥∃𝑦 ∈ 𝐴 (𝜑 ∧ 𝑥 = 𝐵) ↔ ∃𝑦 ∈ 𝐴 𝜑)

Syntax hints:   ↔ wb 196   ∧ wa 384   = wceq 1480  ∃wex 1701   ∈ wcel 1992  ∃wrex 2913

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-11 2036  ax-12 2049

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1702  df-sb 1883  df-clab 2613  df-clel 2622  df-rex 2918

This theorem is referenced by: (None)