Theorem bj-rexcom4b 34323

Description: Remove from rexcom4b 3471 dependency on ax-ext 2770 and ax-13 2379 (and on df-or 845, df-cleq 2791, df-nfc 2938, df-v 3443). The hypothesis uses 𝑉 instead of V (see bj-isseti 34318 for the motivation). Use bj-rexcom4bv 34322 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 3214	. 2 ⊢ (∃𝑥∃𝑦 ∈ 𝐴 (𝜑 ∧ 𝑥 = 𝐵) ↔ ∃𝑦 ∈ 𝐴 (𝜑 ∧ ∃𝑥 𝑥 = 𝐵))
2		bj-rexcom4b.1	. . . . 5 ⊢ 𝐵 ∈ 𝑉
3	2	bj-isseti 34318	. . . 4 ⊢ ∃𝑥 𝑥 = 𝐵
4	3	biantru 533	. . 3 ⊢ (𝜑 ↔ (𝜑 ∧ ∃𝑥 𝑥 = 𝐵))
5	4	rexbii 3210	. 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)

Syntax hints:   ↔ wb 209   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2111  ∃wrex 3107

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-11 2158

This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2777  df-clel 2870  df-rex 3112

This theorem is referenced by: (None)