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Theorem iotajust 6011
Description: Soundness justification theorem for df-iota 6012. (Contributed by Andrew Salmon, 29-Jun-2011.)
Assertion
Ref Expression
iotajust {𝑦 ∣ {𝑥𝜑} = {𝑦}} = {𝑧 ∣ {𝑥𝜑} = {𝑧}}
Distinct variable groups:   𝑥,𝑧   𝜑,𝑧   𝜑,𝑦   𝑥,𝑦
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem iotajust
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 sneq 4331 . . . . 5 (𝑦 = 𝑤 → {𝑦} = {𝑤})
21eqeq2d 2770 . . . 4 (𝑦 = 𝑤 → ({𝑥𝜑} = {𝑦} ↔ {𝑥𝜑} = {𝑤}))
32cbvabv 2885 . . 3 {𝑦 ∣ {𝑥𝜑} = {𝑦}} = {𝑤 ∣ {𝑥𝜑} = {𝑤}}
4 sneq 4331 . . . . 5 (𝑤 = 𝑧 → {𝑤} = {𝑧})
54eqeq2d 2770 . . . 4 (𝑤 = 𝑧 → ({𝑥𝜑} = {𝑤} ↔ {𝑥𝜑} = {𝑧}))
65cbvabv 2885 . . 3 {𝑤 ∣ {𝑥𝜑} = {𝑤}} = {𝑧 ∣ {𝑥𝜑} = {𝑧}}
73, 6eqtri 2782 . 2 {𝑦 ∣ {𝑥𝜑} = {𝑦}} = {𝑧 ∣ {𝑥𝜑} = {𝑧}}
87unieqi 4597 1 {𝑦 ∣ {𝑥𝜑} = {𝑦}} = {𝑧 ∣ {𝑥𝜑} = {𝑧}}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1632  {cab 2746  {csn 4321   cuni 4588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-rex 3056  df-sn 4322  df-uni 4589
This theorem is referenced by: (None)
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