Theorem bj-rexcom4b 37577

Description: Remove from rexcom4b 3488 dependency on ax-ext 2737 and ax-13 2406 (and on df-or 862, df-cleq 2757, df-nfc 2914, df-v 3459). The hypothesis uses 𝑉 instead of V (see bj-isseti 37572 for the motivation). Use bj-rexcom4bv 37576 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		rexcom4a 3297	. 2 ⊢ (∃𝑥∃𝑦 ∈ 𝐴 (𝜑 ∧ 𝑥 = 𝐵) ↔ ∃𝑦 ∈ 𝐴 (𝜑 ∧ ∃𝑥 𝑥 = 𝐵))
2		bj-rexcom4b.1	. . . . 5 ⊢ 𝐵 ∈ 𝑉
3	2	bj-isseti 37572	. . . 4 ⊢ ∃𝑥 𝑥 = 𝐵
4	3	biantru 539	. . 3 ⊢ (𝜑 ↔ (𝜑 ∧ ∃𝑥 𝑥 = 𝐵))
5	4	rexbii 3114	. 2 ⊢ (∃𝑦 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 (𝜑 ∧ ∃𝑥 𝑥 = 𝐵))
6	1, 5	bitr4i 281	1 ⊢ (∃𝑥∃𝑦 ∈ 𝐴 (𝜑 ∧ 𝑥 = 𝐵) ↔ ∃𝑦 ∈ 𝐴 𝜑)

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2146  ∃wrex 3091

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-11 2195

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2744  df-clel 2840  df-rex 3092

This theorem is used by: (None)