Theorem iotavalsb 45034

Description: Theorem *14.242 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 11-Jul-2011.)

Assertion

Ref	Expression
iotavalsb	⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ([𝑦 / 𝑧]𝜓 ↔ [(℩𝑥𝜑) / 𝑧]𝜓))

Distinct variable groups:   𝑥,𝑦   𝜑,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑧)   𝜓(𝑥,𝑦,𝑧)

Proof of Theorem iotavalsb

Step	Hyp	Ref	Expression
1		19.8a 2223	. 2 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦))
2		eu6 2608	. . 3 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦))
3		iotavalb 45031	. . . 4 ⊢ (∃!𝑥𝜑 → (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ↔ (℩𝑥𝜑) = 𝑦))
4		dfsbcq 3755	. . . . 5 ⊢ (𝑦 = (℩𝑥𝜑) → ([𝑦 / 𝑧]𝜓 ↔ [(℩𝑥𝜑) / 𝑧]𝜓))
5	4	eqcoms 2777	. . . 4 ⊢ ((℩𝑥𝜑) = 𝑦 → ([𝑦 / 𝑧]𝜓 ↔ [(℩𝑥𝜑) / 𝑧]𝜓))
6	3, 5	biimtrdi 256	. . 3 ⊢ (∃!𝑥𝜑 → (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ([𝑦 / 𝑧]𝜓 ↔ [(℩𝑥𝜑) / 𝑧]𝜓)))
7	2, 6	sylbir 238	. 2 ⊢ (∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ([𝑦 / 𝑧]𝜓 ↔ [(℩𝑥𝜑) / 𝑧]𝜓)))
8	1, 7	mpcom 39	1 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ([𝑦 / 𝑧]𝜓 ↔ [(℩𝑥𝜑) / 𝑧]𝜓))

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1565   = wceq 1567  ∃wex 1806  ∃!weu 2602  [wsbc 3753  ℩cio 6491

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-12 2219  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1570  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-v 3465  df-sbc 3754  df-un 3918  df-ss 3930  df-sn 4595  df-pr 4597  df-uni 4877  df-iota 6493

This theorem is referenced by: (None)