Theorem isarep1OLD 6639

Description: Obsolete version of isarep1 6638 as of 19-Dec-2024. (Contributed by NM, 26-Oct-2006.) (Proof shortened by Mario Carneiro, 4-Dec-2016.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
isarep1OLD	⊢ (𝑏 ∈ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴) ↔ ∃𝑥 ∈ 𝐴 [𝑏 / 𝑦]𝜑)

Distinct variable groups:   𝑥,𝐴   𝑥,𝑏,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑏)   𝐴(𝑦,𝑏)

Proof of Theorem isarep1OLD

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3479	. . 3 ⊢ 𝑏 ∈ V
2	1	elima 6065	. 2 ⊢ (𝑏 ∈ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴) ↔ ∃𝑧 ∈ 𝐴 𝑧{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑏)
3		df-br 5150	. . . 4 ⊢ (𝑧{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑏 ↔ ⟨𝑧, 𝑏⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
4		opelopabsb 5531	. . . 4 ⊢ (⟨𝑧, 𝑏⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝑧 / 𝑥][𝑏 / 𝑦]𝜑)
5		sbsbc 3782	. . . . . 6 ⊢ ([𝑏 / 𝑦]𝜑 ↔ [𝑏 / 𝑦]𝜑)
6	5	sbbii 2080	. . . . 5 ⊢ ([𝑧 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑧 / 𝑥][𝑏 / 𝑦]𝜑)
7		sbsbc 3782	. . . . 5 ⊢ ([𝑧 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑧 / 𝑥][𝑏 / 𝑦]𝜑)
8	6, 7	bitr2i 276	. . . 4 ⊢ ([𝑧 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑧 / 𝑥][𝑏 / 𝑦]𝜑)
9	3, 4, 8	3bitri 297	. . 3 ⊢ (𝑧{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑏 ↔ [𝑧 / 𝑥][𝑏 / 𝑦]𝜑)
10	9	rexbii 3095	. 2 ⊢ (∃𝑧 ∈ 𝐴 𝑧{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑏 ↔ ∃𝑧 ∈ 𝐴 [𝑧 / 𝑥][𝑏 / 𝑦]𝜑)
11		nfs1v 2154	. . 3 ⊢ Ⅎ𝑥[𝑧 / 𝑥][𝑏 / 𝑦]𝜑
12		nfv 1918	. . 3 ⊢ Ⅎ𝑧[𝑏 / 𝑦]𝜑
13		sbequ12r 2245	. . 3 ⊢ (𝑧 = 𝑥 → ([𝑧 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑏 / 𝑦]𝜑))
14	11, 12, 13	cbvrexw 3305	. 2 ⊢ (∃𝑧 ∈ 𝐴 [𝑧 / 𝑥][𝑏 / 𝑦]𝜑 ↔ ∃𝑥 ∈ 𝐴 [𝑏 / 𝑦]𝜑)
15	2, 10, 14	3bitri 297	1 ⊢ (𝑏 ∈ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴) ↔ ∃𝑥 ∈ 𝐴 [𝑏 / 𝑦]𝜑)

Syntax hints:   ↔ wb 205  [wsb 2068   ∈ wcel 2107  ∃wrex 3071  [wsbc 3778  ⟨cop 4635   class class class wbr 5149  {copab 5211   “ cima 5680

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-xp 5683  df-cnv 5685  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690

This theorem is referenced by: (None)